Question

Use the definition of the derivative to find the derivatives described in Exercises 55-58.

Given$f^{\left(3\right)}\left(x\right)=3x^2+1$find$\frac{d^{4}f}{d{x}^4}$and${\left.\frac{d^{4}f}{d{x}^4}\right|}_2$

Short answer

The value is$12$

Step-by-step solution

Step 1. Given information

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

Use the definition of derivative and calculate

Calculating, we get

$$\begin{gathered} \underset{z\to x}{\lim }\frac{f^{\left(3\right)}\left(z\right)-f^{\left(3\right)}\left(x\right)}{z-x}=\underset{z\to x}{\lim }\frac{3z^2+1-\left(3x^2+1\right)}{z-x} \\ \underset{z\to x}{\lim }\frac{f^{\left(3\right)}\left(z\right)-f^{\left(3\right)}\left(x\right)}{z-x}=\underset{z\to x}{\lim }\frac{3\left(z^2-x^2\right)}{z-x} \\ =\underset{z\to x}{\lim }\frac{3(z-x)(z+x)}{z-x} \\ =\underset{z\to x}{\lim }3(z+x) \\ =3(x+x) \\ =6x \end{gathered}$$

$$\begin{gathered} {\left.\frac{d^{4}f}{d{x}^4}\right|}_{x=2}=6\left(2\right) \\ {\left.\frac{d^{4}f}{d{x}^4}\right|}_{x=2}=12 \end{gathered}$$