Answered: Differentiate each function.…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter4: Exponential And Logarithmic Functions

Section: Chapter Questions

Problem 3CC: If xis large, which function grows faster, f(x)=2x or g(x)=x2?

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Question

Differentiate each function.

a)f(x)=e^9x+2

b)f(x)=10e^[4(x^2)+11]

c) f(x)=25x^3e^2x

Expert Solution

Step 1

Given data:

The expression for the given function is f(x)=e^9x+2.

The expression for the derivative of the given function is,

$f' (x) = \frac{d}{d x} f (x)$

Substitute the given function in the above expression.

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} (e^{9 x} + 2) \\ = & \frac{d}{d x} (e^{9 x}) + \frac{d}{d x} (2) \\ = & 9 e^{9 x} + 0 \\ = & 9 e^{9 x} \end{array}$

Thus, the derivative of the given function is 9e^9x.

Step 2

Given data:

The expression for the given function is f(x)=10e^[4(x^2)+11].

The expression for the derivative of the given function is,

$f' (x) = \frac{d}{d x} f (x)$

Substitute the given function in the above expression.

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} (10 e^{(4 x^{2} + 11)}) \\ = & 10 \frac{d}{d x} (e^{(4 x^{2} + 11)}) \\ = & 10 e^{(4 x^{2} + 11)} \frac{d}{d x} (4 x^{2} + 11) \\ = & 10 e^{(4 x^{2} + 11)} (8 x) \\ = & 80 x e^{(4 x^{2} + 11)} \end{array}$

Thus, the derivative of the given function is 80xe^{(4(x^2)+11)}.

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