Chapter 2: Q. 57 (page 185)

Use the definition of the derivative to find the derivatives described in Exercises 55-58.
Given $f^{(3)} (x) = 3 x^{2} + 1$ find $\frac{d^{4} f}{d x^{4}}$ and ${\frac{d^{4} f}{d x^{4}}|}_{2}$

Short Answer

Expert verified

The value is $12$

Step by step solution

Step 1. Given information

Given function $f^{(3)} (x) = 3 x^{2} + 1$

Use the definition of derivative and calculate

Calculating, we get

$\lim_{z \to x} \frac{f^{(3)} (z) - f^{(3)} (x)}{z - x} = \lim_{z \to x} \frac{3 z^{2} + 1 - (3 x^{2} + 1)}{z - x} \lim_{z \to x} \frac{f^{(3)} (z) - f^{(3)} (x)}{z - x} = \lim_{z \to x} \frac{3 (z^{2} - x^{2})}{z - x} = \lim_{z \to x} \frac{3 (z - x) (z + x)}{z - x} = \lim_{z \to x} 3 (z + x) = 3 (x + x) = 6 x$

${\frac{d^{4} f}{d x^{4}}|}_{x = 2} = 6 (2) {\frac{d^{4} f}{d x^{4}}|}_{x = 2} = 12$

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Use the definition of the derivative to find the derivatives described in Exercises 55-58.
Given $f^{(3)} (x) = 3 x^{2} + 1$ find $\frac{d^{4} f}{d x^{4}}$ and ${\frac{d^{4} f}{d x^{4}}|}_{2}$

Short Answer

Step by step solution

Step 1. Given information

Use the definition of derivative and calculate

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Use the definition of the derivative to find the derivatives described in Exercises 55-58.Givenf(3)(x)=3x2+1findd4fdx4andd4fdx42

Short Answer

Step by step solution

Step 1. Given information

Use the definition of derivative and calculate

One App. One Place for Learning.

Most popular questions from this chapter

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Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Use the definition of the derivative to find the derivatives described in Exercises 55-58.
Given $f^{(3)} (x) = 3 x^{2} + 1$ find $\frac{d^{4} f}{d x^{4}}$ and ${\frac{d^{4} f}{d x^{4}}|}_{2}$