Chapter 3.1, Problem 25E

To determine

To find:The one function which is its own derivative.

Answer to Problem 25E

The required function which is its own derivative is the function (iv).

Explanation of Solution

Given information:

The graphs:

(i) $y=\sin x$ , (ii) $y=x$ , (iii) $y=\sqrt{x}$ , (iv) $y=e^x$ , (v) $y=x^2$

For $x=0$ ;

(i) $\begin{array}{c}y=\sin \left(0\right)\\ =0\end{array}$

(ii) $y=0$

(iii) $\begin{array}{c}y=\sqrt{0}\\ =0\end{array}$

It is seen that in the functions (i), (ii) and (iii) the graphs have nonzero slopes at $x=0$ .

This means none of the above graphs are their own derivative.

For the function (v);

Substitute 1 for x in the function $y=x^2$ .

  $\begin{array}{c}y={\left(1\right)}^2\\ =1\end{array}$

Differentiate the above function.

  $y^{\prime }=2x$

Substitute 1 for x in the above function.

  $\begin{array}{c}{y}^{\prime }=2\left(1\right)\\ =2\end{array}$

Observe that the graph of the function (v) is not its own derivative as y = 1 and $y^{\prime }=2$ at $x=1$ .

This means the function $y=e^x$ is its own derivative.

Hence, the required function which is its own derivative is the function (iv).