Chapter 12.3, Problem 77E

(a)

To determine

To sketch: the graphs of the functions $f\text{ and }{f}^{\prime }$ on same set of coordinate axes.

Explanation of Solution

Given information:

The given function $f\left(x\right)=x^2$ .

Calculation:

The derivative of the function $f\left(x\right)=x^2$ is computed as follows,

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}\left(x^2\right)\\ =2x^{2-1}\left[\because \frac{d}{dx}\left(x^n\right)=n{x}^{n-1}\right]\\ =2x\end{array}$

The derivative of the function $f\left(x\right)=x^2$ is 2x .

Graph:

The graph of the functions $f\text{ and }{f}^{\prime }$ are shown in Figure 1.

  

Interpretation:

From Figure 1, it is observed that the shape of the function f is parabola and the shape of the function $f^{\prime }\left(x\right)=2x$ is a straight line. also, the value of the slope of the tangent at $\left(x,f\left(x\right)\right)$ is same as the value of $f^{\prime }\left(x\right)$ .

(b)

To determine

To sketch: the graphs of the functions $g\text{ and }{g}^{\prime }$ on same set of coordinate axes.

Explanation of Solution

Given information:

The given function $g\left(x\right)=x^3$ .

Calculation:

The derivative of the function $g\left(x\right)=x^3$ is computed as follows,

  $\begin{array}{c}{g}^{\prime }\left(x\right)=\frac{d}{dx}\left(x^3\right)\\ =3x^{3-1}\left[\because \frac{d}{dx}\left(x^n\right)=n{x}^{n-1}\right]\\ =3x^2\end{array}$

The derivative of the function g is $3x^2$ .

Graph:

The graph of the functions $g\text{ and }{g}^{\prime }$ are shown in Figure 2.

  

Interpretation:

From Figure 2, it is observed that the value of the slope of the tangent at $\left(x,g\left(x\right)\right)$ is same as the value of $g^{\prime }\left(x\right)$ .

(c)

To determine

To identify: the functions $f\text{ and }g$ and its derivatives, use the pattern to make the derivative of the $h\left(x\right)=x^n,n\ge 2$ .

Answer to Problem 77E

The derivative of $h\left(x\right)=x^n$ is $n{x}^{n-1}$ .

Explanation of Solution

Interpretation:

From Figure 1 and From Figure 2, it is observed that the shape of the derivative of the $g\left(x\right)$ is a parabola open upward and the shape of the $f\left(x\right)$ also a parabola open upward but not identically same. the vertex of the both parabolas is same

Thus is, the derivative of $g\left(x\right)$ is constant multiple of the function $f\left(x\right)$ .

Use this pattern to conclude the function $h\left(x\right)=x^n$ .

The derivative of the function $h\left(x\right)=x^n$ is constant multiplication of $x^{n-1}$ .

Therefore, the derivative of $h\left(x\right)=x^n$ is $n{x}^{n-1}$ .