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Q.35

Expert-verified

Calculus

Found in: Page 198

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use the differentiation rules developed in this section to find
the derivatives of the functions
$f (x) = 4 - x^{7}$

The derivative of $f (x) = 4 - x^{7} is - 7 x^{6}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1. Given information

Here function $f (x) = 4 - x^{7}$ is given.

we have to find out the derivatives of this function

Step2. finding the derivatives of the function

By using the power rule, we can solve this problem.

Power rule: For any nonzero rational number k,

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

Thus using this power rule when we do the differentiation , we get

$\frac{d}{d x} (4 - x^{7}) = \frac{d (4)}{d x} - \frac{d (x^{7})}{d x} = 0 - 7 x^{6} since derivative of a constant is 0 = - 7 x^{6} thus the derivative of given function is - 7 x^{6}$

Most popular questions for Math Textbooks

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = \frac{x^{2} - \sqrt[3]{x} + 5 x^{9}}{\sqrt{x^{- 1}}}$

Use the definition of the derivative to find $f$ for each function $f$ in Exercises $39 - 54$

$f (x) = \frac{1}{x^{2} - 1}$

Suppose f is a polynomial of degree n and let k be some integer with $0 \leq k \leq n$ . Prove that if f(x) is of the form $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . . + a_{1} x + a_{0}$

Then $a_{k} = \frac{f^{k} (0)}{k!}$ where $f^{k} (x)$ is the k-th derivative of $f (x) a n d k! = k \cdot (k - 1) \cdot (k - 2) . . . . 2 \cdot 1$

Suppose f is ant cubic polynomial function $f (x) = a x^{3} + b x^{2} + c x + d$ prove that coefficients of f a, b, c, d can be expressed in terms of values of f(x) and its derivatives at the point x=2

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59

role="math" localid="1648284617718" $f (x) = \sqrt{2 x + 1}$

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