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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Follow the instructions in each problem. Show supporting work, not just a final answer, to receive credit on a problem. 1. Given the function (1) below, determine the derivative l'(). Explain by stating where you are using the product rule, quotient rule, and/or known derivatives of exponential and/or trigonometric functions. Please explicitly state which rules) of differentiation you use in determining the derivative.

F(x) = 4x^3 +e^2

Expert Solution

Step 1

Given :

$F (x) = 4 x^{3} + e^{2}$

Now, To find its derivative we would first do $\frac{d y}{d x}$ ;

$\frac{d}{d x} F (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (e^{2})$

Since,

Derivative of a constant is always zero.

For example, if a is any constant , then $\frac{d}{d x} (a) = 0$

Similarly , in the given expression:

$e^{2}$ is a constant exponential form , as it doesn't have x in it ,

So, $\frac{d}{d x} (e^{2}) = 0$ Since, as per the rule of differentiation , $\frac{d}{d x} (a) = 0$

Step 2

Now,

$\frac{d}{d x} F (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (e^{2})$

$\frac{d}{d x} F (x) = 4 \frac{d}{d x} (x^{3}) + 0$ ....... Since $\frac{d}{d x} (e^{2}) = 0$

Now,

Another rule of differentiation is :

$\frac{d}{d x} (x^{n}) = n x^{n - 1}$

Similarly ,

$\frac{d}{d x} (x^{3}) = 3 x^{2}$

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