Chapter 11.3, Problem 24E

a.

To determine

To calculate: The derivative formula of the function, $f\left(x\right)=x^3+1$ and hence find the derivative at the point $\left(-1,0\right)$ .

Answer to Problem 24E

The derivative formula of the function, $f\left(x\right)=x^3+1$ is $f\text{'}\left(x\right)=3x^2$ and the derivative at the point $\left(-1,0\right)$ is 3.

Explanation of Solution

Given information: The function and point are $f\left(x\right)=x^3+1$ and $\left(-1,0\right)$ respectively.

Formula used: The derivative of the function $f\left(x\right)$ is given by, $f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$ such that the limit exists.

Calculation: Find the derivative of the function, $f\left(x\right)=x^3+1$ .

  $\begin{array}{c}f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{\left[{\left(x+h\right)}^3+1\right]-\left(x^3+1\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\left(x^3+h^3+3x^{2}h+3x{h}^2+1\right)-\left(x^3+1\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{h\left(h^2+3x^2+3xh\right)}{h}\\ =\underset{h\to 0}{\lim }{h}^2+3x^2+3xh\end{array}$

Find the value of the limit when h approaches to 0.

  $\begin{array}{c}f\text{'}\left(x\right)={\left(0\right)}^2+3x^2+3x\left(0\right)\\ =3x^2\end{array}$

Thus, the derivative formula is $f\text{'}\left(x\right)=3x^2$ .

Substitute $x=-1$ in the derivative formula.

  $\begin{array}{c}f\text{'}\left(-1\right)=3{\left(-1\right)}^2\\ =3\left(1\right)\\ =3\end{array}$

Thus, the derivative of the function is 3.

b.

To determine

To calculate: The derivative of the function, $f\left(x\right)=x^3+1$ at the point $\left(2,9\right)$ .

Answer to Problem 24E

The derivative of the function, $f\left(x\right)=x^3+1$ at the point $\left(2,9\right)$ is 12.

Explanation of Solution

Given: The function and point are $f\left(x\right)=x^3+1$ and $\left(2,9\right)$ respectively.

Calculation: Substitute $x=2$ in the derivative formula, $f\text{'}\left(x\right)=3x^2$ .

  $\begin{array}{c}f\text{'}\left(12\right)=3{\left(2\right)}^2\\ =3\left(4\right)\\ =12\end{array}$

Thus, the derivative of the function is 12.