4. Given data of f(x) = ln(e" +2) in the following table -1 -0.5 0.5 f(x) | 0.86199480 0.95802009 | f'(x) | 0.15536240 | 0.23269654 | 0.33333333 0.45186776 1.0986123 1.2943768 (i) Construct the Lagrange interpolating polynomial of degree three and use it to approximate f(0.25); (ii) Construct the Hermite interpolating polynomial of degree five (through using xo = -0.5, x1 = 0 and x2 = 0.5) and use it to approximate f(0.25). %3D

Q: f(x) = (x² + In x)(x³ + 4x + 5) f(x) = (x2+5x) In x (Vx-6x)e* %3D

A: We have to find the derivative for each of the functions. 1. fx=x2+lnxx3+4x+52. f(x)=x2+5xlnxx-6xex

Q: Differentiate. 1 F(x) = 2x - 5 F'(x) =O

A: Given the functionf(x)=12x−5

Q: Let ƒ(x) = ln(−5x² + 3x + 10) ƒ'(x) =

A: use logarithmic differentiation

Q: x + 1 X-7' (a) the average rate of change of f(x) on [x, x + h] Given the function f(x) = (b) the…

A:

Q: If f(1) = -5.4, f'(x) is continuous, and f'(x)dx = 20, what is the value of f(4)?

A:

Q: 1. Find f'(x) for f(x) = ln x + 2e - 3x².

A: The given function fx=ln x+2ex-3x2. We have to find the derivative function f'x.

Q: f(–5) = f(4) = 0; f '(-0.5) = 0, f '(x) 0 for x > -0.5 What is the function with those given…

A: We have to find function.

Q: Find g'(8) given that f(8)= -6, f'(8) = 7, and g(x) = g'(8) = 9x + 5 f(x) (Round your answer to four…

A:

Q: d f' in terms of g'. IGO f(x) = xg(x) %3D f'(x) = 5x* + g'(x) f'(x) = xg(x) + 5x5g'(x) f'(x) =…

A: We have to find following given condition

Q: An automobile company is ready to introduce a new line of cars through a national sales campaign.…

A: Given integrating we get,      , Where C is an arbitrary constant       Given, no sales at the…

Q: if f(x) = x² √In x² x . find the derivative of this function f'(x A. f'(x) = B. f'(x) = c. f'(x) =…

A:

Q: LET 9 (3)=4₁ g (3)=-1 FIND H (3) For H(X) = xg(x) 5x-6 FUNCTION

A: solution is given below..

Q: Which derivative rule(s) was (were) used to find the following derivative? Select ALL that apply.…

A: Given function is fx=3x2+5x4x3. We have to differentiate the given function. Used formula:-…

Q: if f(x) = %79.1 x²√In x² x -. find the derivative of this function f'(x) A. f'(x) = - 1 3:16 1-In x²…

A: The given function is; f(x)=x2 sqrt(ln x2)/x.

Q: Find f’(e) of the following function. f(x)= x3 *(ln x- 1/3)

A:

Q: Evaluate h'(4) where h(x) = f(x) · g(x) given the following. • f(4) = 7 • f '(4) = −2.5…

A: Given h( x ) = f( x ).g( x ) as we know that ddx( u.v ) = udvdx + vdudxGiven f( 4 ) = 7 f'( 4 ) =…

Q: Suppose that the graph of f(x) is given below. Which of the following is the smallest?

A: Given that:

Q: 2 Compute the derivative f'(x) of the logistic sigmoi 1 f (x) : 1+ exp(-x)

A: We have to find

Q: f f(x) = x²√Inx² x . find the derivative of this function f'(x) 1-In x² x²√√In x² A. f'(x) = B.…

A:

Q: 6. Find f '(x) for f(x) = %3D 6. f '(x) =

A:

Q: x/f(x) 10. Given h(x) If f(2) = 9, f'(2) = 4, g(2) = -1, g'(2) = -3, calculate h'(2). g² (x)

A:

Q: To the nearest tenth, what is f′(k) if f(x)=0.09e^xln(x) and k=8?

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.