Chapter 14: Q. 27 (page 897)

In Problems 21–32, find the derivative of each function at the given number.
$f (x) = x^{3} + 4 x$ at $- 1$

Short Answer

Expert verified

The derivative of given function at $- 1$ is $7$ .

Step by step solution

Step 1. Given Information

We are given a function,

$f (x) = x^{3} + 4 x$

We have to find the derivative of the given function at $- 1$ .

Step 2. Definition of derivative

The derivative of the function at c is defined as,

$f' (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$

Here, $c = - 1$

$f (x) = x^{3} + 4 x f (- 1) = (- 1)^{3} + 4 (- 1) = - 1 - 4 = - 5$

Step 3. Finding the derivative

By substituting the given function in the derivative definition, we get

$f' (- 1) = \lim_{x \to - 1} \frac{f (x) - f (- 1)}{x - (- 1)} = \lim_{x \to - 1} \frac{x^{3} + 4 x - (- 5)}{x + 1} = \lim_{x \to - 1} \frac{x^{3} + 4 x + 5}{x + 1} = \lim_{x \to - 1} \frac{(x^{2} - x + 5) (x + 1)}{x + 1} = \lim_{x \to - 1} (x^{2} - x + 5) = (- 1)^{2} - (- 1) + 5 = 1 + 1 + 5 = 7$

Hence, the derivative of the function at $- 1$ is $7$ .

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In Problems 21–32, find the derivative of each function at the given number.
$f (x) = x^{3} + 4 x$ at $- 1$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Definition of derivative

Step 3. Finding the derivative

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In Problems 21–32, find the derivative of each function at the given number.f(x)=x3+4xat-1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Definition of derivative

Step 3. Finding the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Calculus

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 21–32, find the derivative of each function at the given number.
$f (x) = x^{3} + 4 x$ at $- 1$