Single Variable Calculus: Early Transcendentals
Single Variable Calculus: Early Transcendentals
8th Edition
ISBN: 9781305270336
Author: James Stewart
Publisher: Cengage Learning
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Chapter 3.4, Problem 1E

Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx.

y = 1 + 4 x 3

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Chapter 3 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 3.1 - (a) How is the number e defined? (b) Use a...Ch. 3.1 - Prob. 2ECh. 3.1 - Differentiate the function. f(x) = 240Ch. 3.1 - Differentiate the function. f(x) = e5Ch. 3.1 - Differentiate the function. f(x) = 5.2x + 2.3Ch. 3.1 - Prob. 6ECh. 3.1 - Prob. 7ECh. 3.1 - Differentiate the function. f(t) = 1.4t5 2.5t2+...Ch. 3.1 - Prob. 9ECh. 3.1 - Prob. 10E
Ch. 3.1 - Prob. 11ECh. 3.1 - Differentiate the function. B(y) = cy6Ch. 3.1 - Prob. 13ECh. 3.1 - Differentiate the function. y = x5/3 x2/3Ch. 3.1 - Differentiate the function. R(a) = (3a + 1)2Ch. 3.1 - Differentiate the function. h(t)=t44etCh. 3.1 - Differentiate the function. S(p)=ppCh. 3.1 - Differentiate the function. y=x3(2+x)Ch. 3.1 - Differentiate the function. y=3ex+4x3Ch. 3.1 - Differentiate the function. S(R) = 4R2Ch. 3.1 - Prob. 21ECh. 3.1 - Prob. 22ECh. 3.1 - Prob. 23ECh. 3.1 - Prob. 24ECh. 3.1 - Prob. 25ECh. 3.1 - Prob. 26ECh. 3.1 - Prob. 27ECh. 3.1 - Prob. 28ECh. 3.1 - Prob. 29ECh. 3.1 - Differentiate the function. D(t)=1+16t2(4t)3Ch. 3.1 - Prob. 31ECh. 3.1 - Differentiate the function. y = ex + 1 + 1Ch. 3.1 - Prob. 33ECh. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Prob. 36ECh. 3.1 - Find equations of the tangent line and normal line...Ch. 3.1 - Prob. 38ECh. 3.1 - Prob. 39ECh. 3.1 - Prob. 40ECh. 3.1 - Prob. 41ECh. 3.1 - Prob. 42ECh. 3.1 - Prob. 43ECh. 3.1 - Prob. 44ECh. 3.1 - Prob. 45ECh. 3.1 - Prob. 46ECh. 3.1 - Prob. 47ECh. 3.1 - Prob. 48ECh. 3.1 - Prob. 49ECh. 3.1 - Prob. 50ECh. 3.1 - Prob. 51ECh. 3.1 - Prob. 52ECh. 3.1 - Prob. 53ECh. 3.1 - Find the points on the curve y = 2x3 + 3x2 12x +...Ch. 3.1 - Prob. 56ECh. 3.1 - Prob. 57ECh. 3.1 - Prob. 58ECh. 3.1 - Prob. 59ECh. 3.1 - Prob. 60ECh. 3.1 - Prob. 61ECh. 3.1 - Where does the normal line to the parabola y = x2 ...Ch. 3.1 - Draw a diagram to show that there are two tangent...Ch. 3.1 - Prob. 64ECh. 3.1 - Prob. 65ECh. 3.1 - Find the nth derivative of each function by...Ch. 3.1 - Prob. 67ECh. 3.1 - The equation y" + y' 2y = x2 is called a...Ch. 3.1 - Find a cubic function y = ax3 + bx2 + cx + d whose...Ch. 3.1 - Prob. 70ECh. 3.1 - Prob. 71ECh. 3.1 - At what numbers is the following function g...Ch. 3.1 - Prob. 73ECh. 3.1 - Prob. 74ECh. 3.1 - Find the parabola with equation y = ax2 + bx whose...Ch. 3.1 - Prob. 76ECh. 3.1 - Prob. 77ECh. 3.1 - Prob. 78ECh. 3.1 - What is the value of c such that the line y = 2x +...Ch. 3.1 - Prob. 80ECh. 3.1 - Prob. 81ECh. 3.1 - A tangent line is drawn to the hyperbola xy = c at...Ch. 3.1 - Prob. 83ECh. 3.1 - Prob. 84ECh. 3.1 - Prob. 85ECh. 3.1 - Prob. 86ECh. 3.2 - Find the derivative of f(x) = (1 + 2x2)(x x2) in...Ch. 3.2 - Find the derivative o f the function...Ch. 3.2 - Prob. 3ECh. 3.2 - Differentiate. g(x)=(x+22)exCh. 3.2 - Differentiate. y=xexCh. 3.2 - Differentiate. y=ex1exCh. 3.2 - Prob. 7ECh. 3.2 - Differentiate. G(x)=x222x+1Ch. 3.2 - Differentiate. H(u)=(uu)(u+u)Ch. 3.2 - Prob. 10ECh. 3.2 - Prob. 11ECh. 3.2 - Prob. 12ECh. 3.2 - Prob. 13ECh. 3.2 - Prob. 14ECh. 3.2 - Prob. 15ECh. 3.2 - Prob. 16ECh. 3.2 - Prob. 17ECh. 3.2 - Prob. 18ECh. 3.2 - Prob. 19ECh. 3.2 - Prob. 20ECh. 3.2 - Prob. 21ECh. 3.2 - Prob. 22ECh. 3.2 - Prob. 23ECh. 3.2 - Differentiate. F(t)=AtBt2+Ct3Ch. 3.2 - Prob. 25ECh. 3.2 - Differentiate. f(x)=ax+bcx+dCh. 3.2 - Prob. 27ECh. 3.2 - Prob. 28ECh. 3.2 - Prob. 29ECh. 3.2 - Prob. 30ECh. 3.2 - Prob. 31ECh. 3.2 - Prob. 32ECh. 3.2 - Prob. 33ECh. 3.2 - Find equations of the tangent line and normal line...Ch. 3.2 - Prob. 35ECh. 3.2 - Prob. 36ECh. 3.2 - Prob. 37ECh. 3.2 - Prob. 38ECh. 3.2 - Prob. 39ECh. 3.2 - Prob. 40ECh. 3.2 - Prob. 41ECh. 3.2 - Prob. 42ECh. 3.2 - Prob. 43ECh. 3.2 - Prob. 44ECh. 3.2 - Prob. 45ECh. 3.2 - If h(2) = 4 and h'(2) = 3, find ddx(h(x)x)|x=2Ch. 3.2 - Prob. 47ECh. 3.2 - Prob. 48ECh. 3.2 - If f and g are the functions whose graphs are...Ch. 3.2 - Prob. 50ECh. 3.2 - Prob. 51ECh. 3.2 - If f is a differentiable function, find an...Ch. 3.2 - Prob. 53ECh. 3.2 - Find equations of the tangent lines to the curve...Ch. 3.2 - Prob. 55ECh. 3.2 - Prob. 56ECh. 3.2 - Prob. 57ECh. 3.2 - A manufacturer produces bolts of a fabric with a...Ch. 3.2 - Prob. 59ECh. 3.2 - The biomass B(t) of a fish population is the total...Ch. 3.2 - (a) Use the Product Rule twice to prove that if f,...Ch. 3.2 - (a) If F(x) = f(x) g(x), where f and g have...Ch. 3.2 - Find expressions for the first five derivatives of...Ch. 3.2 - Prob. 64ECh. 3.3 - Prob. 1ECh. 3.3 - Differentiate. f(x) = x cos x + 2 tan xCh. 3.3 - Differentiate. f(x) = ex cos xCh. 3.3 - Differentiate. y = 2 sec x csc xCh. 3.3 - Differentiate. y = sec tanCh. 3.3 - Prob. 6ECh. 3.3 - Prob. 7ECh. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Prob. 10ECh. 3.3 - Differentiate f()=sin1+cosCh. 3.3 - Prob. 12ECh. 3.3 - Prob. 13ECh. 3.3 - Differentiate. y=sint1+tantCh. 3.3 - Differentiate. f() = cos sinCh. 3.3 - Differentiate. f(t) = tet cot tCh. 3.3 - Prob. 17ECh. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - Prove, using the definition of derivative. that if...Ch. 3.3 - Prob. 21ECh. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Prob. 23ECh. 3.3 - Prob. 24ECh. 3.3 - Prob. 25ECh. 3.3 - Prob. 26ECh. 3.3 - Prob. 27ECh. 3.3 - Prob. 28ECh. 3.3 - If H() = sin , find H'() and H"( ).Ch. 3.3 - If f(t) = sec t, find f"(/4).Ch. 3.3 - Prob. 31ECh. 3.3 - Prob. 32ECh. 3.3 - For what values of x does the graph of f have a...Ch. 3.3 - Prob. 34ECh. 3.3 - Prob. 35ECh. 3.3 - An elastic band is hung on a hook and a mass is...Ch. 3.3 - Prob. 37ECh. 3.3 - Prob. 38ECh. 3.3 - Prob. 39ECh. 3.3 - Prob. 40ECh. 3.3 - Prob. 41ECh. 3.3 - Prob. 42ECh. 3.3 - Prob. 43ECh. 3.3 - Prob. 44ECh. 3.3 - Prob. 45ECh. 3.3 - Prob. 46ECh. 3.3 - Prob. 47ECh. 3.3 - Prob. 48ECh. 3.3 - Prob. 49ECh. 3.3 - Prob. 50ECh. 3.3 - Find the given derivative by finding the first few...Ch. 3.3 - Find the given derivative by finding the first few...Ch. 3.3 - Prob. 53ECh. 3.3 - Prob. 54ECh. 3.3 - Differentiate each trigonometric identity to...Ch. 3.3 - A semicircle with diameter PQ sits on an isosceles...Ch. 3.3 - The figure shows a circular arc of length s and a...Ch. 3.3 - Prob. 58ECh. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 3ECh. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 5ECh. 3.4 - Prob. 6ECh. 3.4 - Find the derivative of the function. F(x) = (5x6 +...Ch. 3.4 - Find the derivative of the function. F (x) = (1 +...Ch. 3.4 - Find the derivative of the function. f(x)=5x+1Ch. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.4 - Prob. 13ECh. 3.4 - Find the derivative of the function. f(t) = t sin ...Ch. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.4 - Find the derivative of the function. f(x) = (2x ...Ch. 3.4 - Find the derivative of the function. g(x) = (x2 +...Ch. 3.4 - Prob. 19ECh. 3.4 - Find the derivative of the function. F(t) = (3t ...Ch. 3.4 - Prob. 21ECh. 3.4 - Find the derivative of the function. y=(x+1x)5Ch. 3.4 - Prob. 23ECh. 3.4 - Find the derivative of the function. f(t)2t3Ch. 3.4 - Prob. 25ECh. 3.4 - Prob. 26ECh. 3.4 - Prob. 27ECh. 3.4 - Find the derivative of the function. f(z) =...Ch. 3.4 - Prob. 29ECh. 3.4 - Prob. 30ECh. 3.4 - Prob. 31ECh. 3.4 - Prob. 32ECh. 3.4 - Prob. 33ECh. 3.4 - Prob. 34ECh. 3.4 - Prob. 35ECh. 3.4 - Find the derivative of the function. y = x2 e1/xCh. 3.4 - Prob. 37ECh. 3.4 - Prob. 38ECh. 3.4 - Prob. 39ECh. 3.4 - Prob. 40ECh. 3.4 - Prob. 41ECh. 3.4 - Prob. 42ECh. 3.4 - Find the derivative of the function. g(x) = (2...Ch. 3.4 - Prob. 44ECh. 3.4 - Prob. 45ECh. 3.4 - Prob. 46ECh. 3.4 - Prob. 47ECh. 3.4 - Prob. 48ECh. 3.4 - Prob. 49ECh. 3.4 - Find y and y. y=eexCh. 3.4 - Find an equation of the tangent line to the curve...Ch. 3.4 - Find an equation of the tangent line to the curve...Ch. 3.4 - Prob. 53ECh. 3.4 - Prob. 54ECh. 3.4 - Prob. 55ECh. 3.4 - Prob. 56ECh. 3.4 - Prob. 57ECh. 3.4 - Prob. 58ECh. 3.4 - Prob. 59ECh. 3.4 - At what point on the curve y=1+2x is the tangent...Ch. 3.4 - Prob. 61ECh. 3.4 - Prob. 62ECh. 3.4 - A table of values for f, g, f, and g is given. (a)...Ch. 3.4 - Let f and g be the functions in Exercise 63. (a)...Ch. 3.4 - Prob. 65ECh. 3.4 - Prob. 66ECh. 3.4 - Prob. 67ECh. 3.4 - Suppose f is differentiable on and is a real...Ch. 3.4 - Suppose f is differentiable on . Let F(x) = f(ex)...Ch. 3.4 - Prob. 70ECh. 3.4 - Prob. 71ECh. 3.4 - Prob. 72ECh. 3.4 - Prob. 73ECh. 3.4 - Prob. 74ECh. 3.4 - Prob. 75ECh. 3.4 - Prob. 76ECh. 3.4 - Prob. 77ECh. 3.4 - Find the 1000th derivative of f(x) = xex.Ch. 3.4 - The displacement of a particle on a vibrating...Ch. 3.4 - If the equation of motion of a particle is given...Ch. 3.4 - Prob. 81ECh. 3.4 - Prob. 82ECh. 3.4 - The motion of a spring that is subject to a...Ch. 3.4 - Prob. 84ECh. 3.4 - The average blood alcohol concentration (BAC) of...Ch. 3.4 - In Section 1.4 we modeled the world population...Ch. 3.4 - Prob. 87ECh. 3.4 - Prob. 88ECh. 3.4 - Prob. 89ECh. 3.4 - The table gives the US population from 1790 to...Ch. 3.4 - Prob. 93ECh. 3.4 - Prob. 94ECh. 3.4 - (a) If n is a positive integer, prove that...Ch. 3.4 - Prob. 96ECh. 3.4 - Prob. 97ECh. 3.4 - Prob. 98ECh. 3.4 - Prob. 99ECh. 3.4 - If y = f(u) and u = g(x), where f and g possess...Ch. 3.5 - (a) Find y by implicit differentiation. (b) Solve...Ch. 3.5 - Prob. 2ECh. 3.5 - Prob. 3ECh. 3.5 - Prob. 4ECh. 3.5 - Find dy/dx by implicit differentiation. 5. x2 4xy...Ch. 3.5 - Prob. 6ECh. 3.5 - Find dy/dx by implicit differentiation. 7. x4 +...Ch. 3.5 - Find dy/dx by implicit differentiation. 8. x3 xy2...Ch. 3.5 - Find dy/dx by implicit differentiation. 9....Ch. 3.5 - Find dy/dx by implicit differentiation. 10. xey =...Ch. 3.5 - Find dy/dx by implicit differentiation. 11. y cos...Ch. 3.5 - Find dy/dx by implicit differentiation. 12....Ch. 3.5 - Find dy/dx by implicit differentiation. 13....Ch. 3.5 - Prob. 14ECh. 3.5 - Prob. 15ECh. 3.5 - Prob. 16ECh. 3.5 - Prob. 17ECh. 3.5 - Prob. 18ECh. 3.5 - Find dy/dx by implicit differentiation. 19....Ch. 3.5 - Prob. 20ECh. 3.5 - Prob. 21ECh. 3.5 - Prob. 22ECh. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Prob. 25ECh. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 27ECh. 3.5 - Prob. 28ECh. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 30ECh. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 32ECh. 3.5 - (a) The curve with equation y2 = 5x4 x2 is called...Ch. 3.5 - Prob. 34ECh. 3.5 - Find y by implicit differentiation. 35. x2 + 4y2 =...Ch. 3.5 - Prob. 36ECh. 3.5 - Prob. 37ECh. 3.5 - Prob. 38ECh. 3.5 - Prob. 39ECh. 3.5 - If x2 + xy + y3 = 1, find the value of y at the...Ch. 3.5 - Prob. 43ECh. 3.5 - Prob. 44ECh. 3.5 - Find an equation of the tangent line to the...Ch. 3.5 - Prob. 46ECh. 3.5 - Show, using implicit differentiation, that any...Ch. 3.5 - Prob. 48ECh. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Prob. 52ECh. 3.5 - Prob. 53ECh. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Prob. 55ECh. 3.5 - Prob. 56ECh. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Prob. 58ECh. 3.5 - Prob. 59ECh. 3.5 - Prob. 60ECh. 3.5 - Prob. 61ECh. 3.5 - Prob. 62ECh. 3.5 - Prove the formula for (d/dx)(cos1x) by the same...Ch. 3.5 - (a) One way of defining sec1x is to say that...Ch. 3.5 - Prob. 65ECh. 3.5 - Prob. 66ECh. 3.5 - Prob. 67ECh. 3.5 - Two curves are orthogonal if their tangent lines...Ch. 3.5 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 3.5 - Prob. 70ECh. 3.5 - Prob. 71ECh. 3.5 - Prob. 73ECh. 3.5 - (a) Where does the normal line to the ellipse x2 ...Ch. 3.5 - Prob. 75ECh. 3.5 - Prob. 76ECh. 3.5 - Prob. 77ECh. 3.5 - Prob. 78ECh. 3.5 - The Bessel function of order 0, y = J(x),...Ch. 3.5 - The figure shows a lamp located three units to the...Ch. 3.6 - Explain why the natural logarithmic function y =...Ch. 3.6 - Differentiate the function. f(x) = x ln x xCh. 3.6 - Differentiate the function. f(x ) = sin(ln x)Ch. 3.6 - Differentiate the function. f(x) = ln(sin2x)Ch. 3.6 - Differentiate the function. f(x)=ln1xCh. 3.6 - Differentiate the function. y=1lnxCh. 3.6 - Prob. 7ECh. 3.6 - Prob. 8ECh. 3.6 - Prob. 9ECh. 3.6 - Prob. 10ECh. 3.6 - Differentiate the function. F(t) =(ln t)2 sin tCh. 3.6 - Differentiate the function. h(x)=ln(x+x21)Ch. 3.6 - Differentiate the function. G(y)=ln(2y+1)5y2+1Ch. 3.6 - Prob. 14ECh. 3.6 - Differentiate the function. F(s) = ln ln sCh. 3.6 - Differentiate the function. y = ln |1 + t t3|Ch. 3.6 - Differentiate the function. T(z) = 2z log2zCh. 3.6 - Prob. 18ECh. 3.6 - Prob. 19ECh. 3.6 - Differentiate the function. H(z)=a2z2a2+z2Ch. 3.6 - Prob. 21ECh. 3.6 - Prob. 22ECh. 3.6 - Prob. 23ECh. 3.6 - Find y and y. y=lnx1+lnxCh. 3.6 - Prob. 25ECh. 3.6 - Prob. 26ECh. 3.6 - Prob. 27ECh. 3.6 - Prob. 28ECh. 3.6 - Prob. 29ECh. 3.6 - Prob. 30ECh. 3.6 - Prob. 31ECh. 3.6 - Prob. 32ECh. 3.6 - Prob. 33ECh. 3.6 - Prob. 34ECh. 3.6 - Prob. 35ECh. 3.6 - Prob. 36ECh. 3.6 - Prob. 37ECh. 3.6 - Prob. 38ECh. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Prob. 40ECh. 3.6 - Prob. 41ECh. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Prob. 44ECh. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Prob. 46ECh. 3.6 - Prob. 47ECh. 3.6 - Prob. 48ECh. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Prob. 50ECh. 3.6 - Prob. 51ECh. 3.6 - Prob. 52ECh. 3.6 - Prob. 53ECh. 3.6 - Prob. 54ECh. 3.6 - Prob. 55ECh. 3.6 - Prob. 56ECh. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - Prob. 4ECh. 3.7 - Prob. 5ECh. 3.7 - Prob. 6ECh. 3.7 - Prob. 7ECh. 3.7 - Prob. 8ECh. 3.7 - Prob. 9ECh. 3.7 - Prob. 10ECh. 3.7 - Prob. 11ECh. 3.7 - Prob. 12ECh. 3.7 - Prob. 13ECh. 3.7 - Prob. 14ECh. 3.7 - Prob. 15ECh. 3.7 - (a) The volume of a growing spherical cell is...Ch. 3.7 - Prob. 17ECh. 3.7 - Prob. 18ECh. 3.7 - The quantity of charge Q in coulombs (C) that has...Ch. 3.7 - Prob. 20ECh. 3.7 - Prob. 21ECh. 3.7 - Prob. 22ECh. 3.7 - Boyles Law states that when a sample of gas is...Ch. 3.7 - Prob. 24ECh. 3.7 - Prob. 25ECh. 3.7 - Prob. 26ECh. 3.7 - The table shows how the average age of first...Ch. 3.7 - Refer to the law of laminar flow given in Example...Ch. 3.7 - Prob. 30ECh. 3.7 - Prob. 31ECh. 3.7 - The cost function for a certain commodity is C(q)...Ch. 3.7 - Prob. 33ECh. 3.7 - Prob. 34ECh. 3.7 - Patients undergo dialysis treatment to remove urea...Ch. 3.7 - Invasive species often display a wave of advance...Ch. 3.7 - Prob. 37ECh. 3.7 - Prob. 38ECh. 3.7 - Prob. 39ECh. 3.8 - A population of protozoa develops with a constant...Ch. 3.8 - A common inhabitant of human intestines is the...Ch. 3.8 - A bacteria culture initially contains 100 cells...Ch. 3.8 - A bacteria culture grows with constant relative...Ch. 3.8 - The table gives estimates of the world population,...Ch. 3.8 - Prob. 6ECh. 3.8 - Prob. 7ECh. 3.8 - Strontium-90 has a half-life of 28 days. (a) A...Ch. 3.8 - Prob. 9ECh. 3.8 - Prob. 10ECh. 3.8 - Prob. 11ECh. 3.8 - Dinosaur fossils are too old to be reliably dated...Ch. 3.8 - Dinosaur fossils are often dated by using an...Ch. 3.8 - Prob. 14ECh. 3.8 - A roast turkey is taken from an oven when its...Ch. 3.8 - In a murder investigation, the temperature of the...Ch. 3.8 - Prob. 17ECh. 3.8 - Prob. 18ECh. 3.8 - Prob. 19ECh. 3.8 - (a) If 1000 is borrowed at 8% interest, find the...Ch. 3.8 - Prob. 21ECh. 3.8 - (a) How long will it take an investment to double...Ch. 3.9 - Prob. 1ECh. 3.9 - (a) If A is the area of a circle with radius r and...Ch. 3.9 - Prob. 3ECh. 3.9 - The length of a rectangle is increasing at a rate...Ch. 3.9 - A cylindrical tank with radius 5 m is being filled...Ch. 3.9 - Prob. 6ECh. 3.9 - The radius of a spherical ball is increasing at a...Ch. 3.9 - Prob. 8ECh. 3.9 - Suppose y=2x+1, where x and y are functions of t....Ch. 3.9 - Prob. 10ECh. 3.9 - If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4,...Ch. 3.9 - A particle is moving along a hyperbola xy = 8. As...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - Two cars start moving from the same point. One...Ch. 3.9 - Prob. 18ECh. 3.9 - A man starts walking north at 4 ft/s from a point...Ch. 3.9 - A baseball diamond is a square with side 90 ft. A...Ch. 3.9 - Prob. 21ECh. 3.9 - A boat is pulled into a dock by a rope attached to...Ch. 3.9 - Prob. 23ECh. 3.9 - Prob. 24ECh. 3.9 - Water is leaking out of an inverted conical tank...Ch. 3.9 - A trough is 10 ft long and its ends have the shape...Ch. 3.9 - A water trough is 10m long and a cross-section has...Ch. 3.9 - Prob. 28ECh. 3.9 - Gravel is being dumped from a conveyor belt at a...Ch. 3.9 - A kite 100ft above the ground moves horizontally...Ch. 3.9 - The sides of an equilateral triangle are...Ch. 3.9 - How fast is the angle between the ladder and the...Ch. 3.9 - Prob. 33ECh. 3.9 - Prob. 34ECh. 3.9 - If the minute hand of a clock has length r (in...Ch. 3.9 - Prob. 36ECh. 3.9 - Boyles Law states that when a sample of gas is...Ch. 3.9 - When air expands adiabatically (without gaining or...Ch. 3.9 - Prob. 39ECh. 3.9 - Prob. 40ECh. 3.9 - Prob. 41ECh. 3.9 - Two carts, A and B, are connected by a rope 39 ft...Ch. 3.9 - Prob. 43ECh. 3.9 - A lighthouse is located on a small island 3 km...Ch. 3.9 - Prob. 45ECh. 3.9 - Prob. 46ECh. 3.9 - Prob. 47ECh. 3.9 - Prob. 48ECh. 3.9 - Prob. 49ECh. 3.9 - Prob. 50ECh. 3.10 - Find the linearization L(x) of the function at n....Ch. 3.10 - Find the linearization L(x) of the function at n....Ch. 3.10 - Find the linearization L(x) of the function at n....Ch. 3.10 - Find the linearization L(x) of the function at n....Ch. 3.10 - Find the linear approximation of the function...Ch. 3.10 - Prob. 6ECh. 3.10 - Prob. 7ECh. 3.10 - Prob. 8ECh. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Find the differential of each function. 11. (a) y...Ch. 3.10 - Prob. 12ECh. 3.10 - Find the differential of each function. 13. (a)...Ch. 3.10 - Prob. 14ECh. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - Prob. 16ECh. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Prob. 20ECh. 3.10 - Prob. 21ECh. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Prob. 24ECh. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Prob. 27ECh. 3.10 - Prob. 28ECh. 3.10 - Prob. 29ECh. 3.10 - Explain, in terms of linear approximations or...Ch. 3.10 - Prob. 31ECh. 3.10 - Prob. 32ECh. 3.10 - Prob. 33ECh. 3.10 - Prob. 34ECh. 3.10 - The circumference of a sphere was measured to be...Ch. 3.10 - Use differentials to estimate the amount of paint...Ch. 3.10 - Prob. 37ECh. 3.10 - Prob. 38ECh. 3.10 - Prob. 39ECh. 3.10 - When blood flows along a blood vessel, the flux F...Ch. 3.10 - Prob. 41ECh. 3.10 - On page 431 of Physics: Calculus, 2d ed., by...Ch. 3.10 - Prob. 43ECh. 3.10 - Prob. 44ECh. 3.11 - Find the numerical value of each expression. 1....Ch. 3.11 - Find the numerical value of each expression. 2....Ch. 3.11 - Prob. 3ECh. 3.11 - Prob. 4ECh. 3.11 - Prob. 5ECh. 3.11 - Prob. 6ECh. 3.11 - Prob. 7ECh. 3.11 - Prob. 8ECh. 3.11 - Prob. 9ECh. 3.11 - Prob. 10ECh. 3.11 - Prob. 11ECh. 3.11 - Prob. 12ECh. 3.11 - Prob. 13ECh. 3.11 - Prob. 14ECh. 3.11 - Prove the identity. 15. sinh 2x = 2 sinh x cosh xCh. 3.11 - Prob. 16ECh. 3.11 - Prob. 17ECh. 3.11 - Prob. 18ECh. 3.11 - Prove the identity. 19. (cosh x + sinh x)n = cosh...Ch. 3.11 - Prob. 20ECh. 3.11 - Prob. 21ECh. 3.11 - Prob. 22ECh. 3.11 - Prob. 23ECh. 3.11 - Prob. 24ECh. 3.11 - Prob. 25ECh. 3.11 - Prob. 26ECh. 3.11 - Prob. 27ECh. 3.11 - Prob. 28ECh. 3.11 - Prob. 29ECh. 3.11 - Prob. 30ECh. 3.11 - Prob. 31ECh. 3.11 - Prob. 32ECh. 3.11 - Prob. 33ECh. 3.11 - Prob. 34ECh. 3.11 - Prob. 35ECh. 3.11 - Find the derivative. Simplify where possible. 36....Ch. 3.11 - Find the derivative. Simplify where possible. 37....Ch. 3.11 - Prob. 38ECh. 3.11 - Prob. 39ECh. 3.11 - Find the derivative. Simplify where possible. 40....Ch. 3.11 - Find the derivative. Simplify where possible. 41....Ch. 3.11 - Prob. 42ECh. 3.11 - Prob. 43ECh. 3.11 - Prob. 44ECh. 3.11 - Prob. 45ECh. 3.11 - Prob. 46ECh. 3.11 - Prob. 47ECh. 3.11 - Prob. 48ECh. 3.11 - Prob. 49ECh. 3.11 - Prob. 50ECh. 3.11 - Prob. 51ECh. 3.11 - Prob. 52ECh. 3.11 - A cable with linear density = 2 kg/m is strung...Ch. 3.11 - A model for the velocity of a falling object after...Ch. 3.11 - Prob. 55ECh. 3.11 - Prob. 56ECh. 3.11 - Prob. 57ECh. 3.11 - Prob. 58ECh. 3 - State each differentiation rule both in symbols...Ch. 3 - Prob. 2RCCCh. 3 - Prob. 3RCCCh. 3 - Prob. 4RCCCh. 3 - Give several examples of how the derivative can be...Ch. 3 - Prob. 6RCCCh. 3 - Prob. 7RCCCh. 3 - Prob. 1RQCh. 3 - Prob. 2RQCh. 3 - Prob. 3RQCh. 3 - Prob. 4RQCh. 3 - Prob. 5RQCh. 3 - Prob. 6RQCh. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 8RQCh. 3 - Prob. 9RQCh. 3 - Prob. 10RQCh. 3 - Prob. 11RQCh. 3 - Prob. 12RQCh. 3 - Prob. 13RQCh. 3 - Prob. 14RQCh. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 1RECh. 3 - Prob. 2RECh. 3 - Prob. 3RECh. 3 - Prob. 4RECh. 3 - Prob. 5RECh. 3 - Prob. 6RECh. 3 - Prob. 7RECh. 3 - Prob. 8RECh. 3 - Prob. 9RECh. 3 - Prob. 10RECh. 3 - Prob. 11RECh. 3 - Calculate y'. 12. y = (arcsin 2x)2Ch. 3 - Prob. 13RECh. 3 - Prob. 14RECh. 3 - Prob. 15RECh. 3 - Prob. 16RECh. 3 - Calculate y'. 17. y=arctanCh. 3 - Prob. 18RECh. 3 - Prob. 19RECh. 3 - Prob. 20RECh. 3 - Prob. 21RECh. 3 - Prob. 22RECh. 3 - Prob. 23RECh. 3 - Prob. 24RECh. 3 - Prob. 25RECh. 3 - Prob. 26RECh. 3 - Prob. 27RECh. 3 - Prob. 28RECh. 3 - Prob. 29RECh. 3 - Calculate y'. 30. y=(x2+1)4(2x+1)3(3x1)5Ch. 3 - Prob. 31RECh. 3 - Prob. 32RECh. 3 - Prob. 33RECh. 3 - Prob. 34RECh. 3 - Prob. 35RECh. 3 - Prob. 36RECh. 3 - Prob. 37RECh. 3 - Prob. 38RECh. 3 - Prob. 39RECh. 3 - Prob. 40RECh. 3 - Prob. 41RECh. 3 - Prob. 42RECh. 3 - Prob. 43RECh. 3 - Prob. 44RECh. 3 - Prob. 45RECh. 3 - Prob. 46RECh. 3 - Prob. 47RECh. 3 - Prob. 48RECh. 3 - Prob. 49RECh. 3 - Prob. 50RECh. 3 - Prob. 51RECh. 3 - Prob. 52RECh. 3 - Prob. 53RECh. 3 - Prob. 54RECh. 3 - Prob. 55RECh. 3 - Prob. 56RECh. 3 - Prob. 57RECh. 3 - Prob. 58RECh. 3 - Prob. 59RECh. 3 - Prob. 60RECh. 3 - Prob. 61RECh. 3 - Prob. 62RECh. 3 - Prob. 63RECh. 3 - Prob. 64RECh. 3 - Prob. 65RECh. 3 - Prob. 66RECh. 3 - Prob. 67RECh. 3 - Prob. 68RECh. 3 - Prob. 69RECh. 3 - Prob. 70RECh. 3 - Prob. 71RECh. 3 - Prob. 72RECh. 3 - Prob. 73RECh. 3 - Prob. 74RECh. 3 - Prob. 75RECh. 3 - Prob. 76RECh. 3 - Prob. 77RECh. 3 - Prob. 78RECh. 3 - Prob. 79RECh. 3 - Prob. 80RECh. 3 - Prob. 81RECh. 3 - Prob. 82RECh. 3 - Prob. 83RECh. 3 - Prob. 84RECh. 3 - Prob. 85RECh. 3 - Prob. 86RECh. 3 - Prob. 87RECh. 3 - A particle moves along a horizontal line so that...Ch. 3 - A particle moves on a vertical line so that its...Ch. 3 - Prob. 90RECh. 3 - Prob. 91RECh. 3 - Prob. 92RECh. 3 - Prob. 93RECh. 3 - Prob. 94RECh. 3 - Prob. 95RECh. 3 - Prob. 96RECh. 3 - Prob. 97RECh. 3 - Prob. 98RECh. 3 - A balloon is rising at a constant speed of 5 ft/s....Ch. 3 - Prob. 100RECh. 3 - Prob. 101RECh. 3 - Prob. 102RECh. 3 - Prob. 103RECh. 3 - Prob. 104RECh. 3 - Prob. 105RECh. 3 - Prob. 106RECh. 3 - Prob. 107RECh. 3 - Prob. 108RECh. 3 - Prob. 109RECh. 3 - Suppose f is a differentiable function such that...Ch. 3 - Prob. 111RECh. 3 - Prob. 112RECh. 3 - Prob. 1PCh. 3 - Prob. 2PCh. 3 - Prob. 3PCh. 3 - Prob. 4PCh. 3 - Prob. 5PCh. 3 - Prob. 6PCh. 3 - Prob. 7PCh. 3 - Prob. 8PCh. 3 - Prob. 9PCh. 3 - Prob. 10PCh. 3 - Prob. 11PCh. 3 - Prob. 12PCh. 3 - Prob. 13PCh. 3 - If f(x)=x46+x45+21+x, calculate f(46)(3). Express...Ch. 3 - The figure shows a rotating wheel with radius 40...Ch. 3 - Prob. 16PCh. 3 - Prob. 17PCh. 3 - Prob. 18PCh. 3 - Let T and N be the tangent and normal lines to the...Ch. 3 - Prob. 20PCh. 3 - Prob. 21PCh. 3 - Let P(x1, y1) be a point on the parabola y2 = 4px...Ch. 3 - Prob. 23PCh. 3 - Prob. 24PCh. 3 - Prob. 25PCh. 3 - Prob. 27PCh. 3 - Prob. 28PCh. 3 - Prob. 29PCh. 3 - Prob. 30PCh. 3 - Find the two points on the curve y = x4 2x2 x...Ch. 3 - Prob. 32PCh. 3 - Prob. 33PCh. 3 - Prob. 34PCh. 3 - Prob. 35P
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Chain Rule dy:dx = dy:du*du:dx; Author: Robert Cappetta;https://www.youtube.com/watch?v=IUYniALwbHs;License: Standard YouTube License, CC-BY
CHAIN RULE Part 1; Author: Btech Maths Hub;https://www.youtube.com/watch?v=TIAw6AJ_5Po;License: Standard YouTube License, CC-BY