Calculus - Standalone book
3rd Edition
ISBN: 9781464125263
Author: Jon Rogawski, Colin Adams
Publisher: W.H. Freeman & Co
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Chapter 7.3, Problem 118E
a.
To determine
To show: the given statement is correct when f and g are two differentiable functions using mathematical methodologies.
b.
To determine
To show: the new proof of product rule by considering f and g are two differentiable functions and given expression using mathematical methodologies.
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Students have asked these similar questions
3. Let f (x) =
(3x2 + 1)?. Find f'(x)in 3 different ways by following the instructions below in parts a, b and c:
a) Algebraically multiply out the expression for f (x) and expand, then take the derivative.
b) View f (x) as (3x? +1)(3x2 + 1) and use the product rule to find f' (x).
C) Apply the chain rule directly to the expression f (x) = (3x² + 1)?.
d) Are your answers in parts a, b, c the same? Why or why not?
8) Find the derivative of the function f(x) =
(2x-1
x² + 1
if x ≤0
if x>0
Previously, we developed the Product Rule and studied how it is employed to differentiate a product of two functions. In particular, recall that if f and g are differentiable functions of x, then
- [ƒ(x) · g(x)] = f(x) · g'(x) + g(x) · f'(x).
d
dx
(a) For each of the following functions, use the Product Rule to find the function's derivative. Notice the label of the derivative (e.g., the derivative of g(x) should be labeled g'(x)).
(i) If g(x) = x sin(x), then g'(x) =
(ii) If h(x) = xe, then
(iii) If p(x) = x ln(x), then
(iv.) If q(x) = x² cos(x), then
(v.) If r(x) = e sin(x), then
(b) Use your work in (a) to help you evaluate the following indefinite integrals. Use differentiation to check your work. (Don't forget the "+C".)
(i) xe²+ eª dx
(ii) [e²(sin(a) + cos(x)) da
(iii) 2x cos(x) - x² sin(x) dx
(iv) x cos(x) + sin(x) dx
=
(v) | 1 + ln(x) dx =
[₁
=
(i) First, observe that
=
9.
=
=
(c) Observe that the examples in (b) work nicely because of the derivatives you were asked to…
Chapter 7 Solutions
Calculus - Standalone book
Ch. 7.1 - Prob. 1PQCh. 7.1 - Prob. 2PQCh. 7.1 - Prob. 3PQCh. 7.1 - Prob. 4PQCh. 7.1 - Prob. 5PQCh. 7.1 - Prob. 6PQCh. 7.1 - Prob. 7PQCh. 7.1 - Prob. 1ECh. 7.1 - Prob. 2ECh. 7.1 - Prob. 3E
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Prob. 56ECh. 7.9 - Prob. 57ECh. 7.9 - Prob. 58ECh. 7.9 - Prob. 59ECh. 7.9 - Prob. 60ECh. 7.9 - Prob. 61ECh. 7.9 - Prob. 62ECh. 7.9 - Prob. 63ECh. 7.9 - Prob. 64ECh. 7.9 - Prob. 65ECh. 7.9 - Prob. 66ECh. 7.9 - Prob. 67ECh. 7.9 - Prob. 68ECh. 7.9 - Prob. 69ECh. 7.9 - Prob. 70ECh. 7.9 - Prob. 71ECh. 7.9 - Prob. 72ECh. 7.9 - Prob. 73ECh. 7.9 - Prob. 74ECh. 7.9 - Prob. 75ECh. 7 - Prob. 1CRECh. 7 - Prob. 2CRECh. 7 - Prob. 3CRECh. 7 - Prob. 4CRECh. 7 - Prob. 5CRECh. 7 - Prob. 6CRECh. 7 - Prob. 7CRECh. 7 - Prob. 8CRECh. 7 - Prob. 9CRECh. 7 - Prob. 10CRECh. 7 - Prob. 11CRECh. 7 - Prob. 12CRECh. 7 - Prob. 13CRECh. 7 - Prob. 14CRECh. 7 - Prob. 15CRECh. 7 - Prob. 16CRECh. 7 - Prob. 17CRECh. 7 - Prob. 18CRECh. 7 - Prob. 19CRECh. 7 - Prob. 20CRECh. 7 - Prob. 21CRECh. 7 - Prob. 22CRECh. 7 - Prob. 23CRECh. 7 - Prob. 24CRECh. 7 - Prob. 25CRECh. 7 - Prob. 26CRECh. 7 - Prob. 27CRECh. 7 - Prob. 28CRECh. 7 - Prob. 29CRECh. 7 - Prob. 30CRECh. 7 - Prob. 31CRECh. 7 - Prob. 32CRECh. 7 - Prob. 33CRECh. 7 - Prob. 34CRECh. 7 - 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- find the function f(x) if f'(x)=(2/1+x2)+(5/7)x5/7 and f(0)=2arrow_forwardWhen two resistors are connected in parallel, the total resistance, R(t), is given by 1 1 1 + R(t) R₁(t) ™R₂(t)' where R₁ (t) and R₂(t) are the resistances of each individual resistor. = (1) (a) Use implicit differentiation, or otherwise, to differentiate Equation (1) to give an ex- dR pression for in terms of R₁ and R₂ and their derivatives. dtarrow_forward2. a) Find and fully factorise the derivative of f(x) = (2x + 1)² using expansion and the power rule. b) Find and fully factorise the derivative of y = (10x + 3)2 using expansion and the power rule. c) Show that the derivative function of f(x) = (9x - 2)² is f'(x) = 18(9x - 2).arrow_forward
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