Chapter 3, Problem 1GYR

To determine

Define the derivative of the function f.

Explain the relation between the domain of function and its derivative with examples.

Explanation of Solution

Consider a function f is differentiable at a domain value a, then $f^{\prime }\left(a\right)$ is a real number. Then, the function $f^{\prime }$ is a new function known as the derivative of the function f. The domain of the derivative function $f^{\prime }$ is the set of all points in the domain of the function f in which the function f itself is differentiable.

Example:

Consider the function $f\left(x\right)=\sqrt{2x}$        (1)

Substitute 0 for x in Equation (1).

$\begin{array}{c}f\left(x\right)=\sqrt{2\times 0}\\ =0\end{array}$

Substitute $\infty$ for x in Equation (1).

$\begin{array}{c}f\left(x\right)=\sqrt{2\times \infty }\\ =\infty \end{array}$

The domain of the function $f\left(x\right)$ is $\underset{\_}{\left[0,\infty \right)}$.

Show the derivative of the function as follows:

Differentiate Equation (1) as follows:

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

Substitute 0 for x in Equation (2).

$\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{1}{\sqrt{0}}\\ =\infty \end{array}$

Substitute $\infty$ for x in Equation (2).

$\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{1}{\sqrt{\infty }}\\ =0\end{array}$

The domain of the function $f^{\prime }\left(x\right)$ is $\underset{\_}{\left(0,\infty \right)}$.

Thus, the domain of the function $f\left(x\right)$ and the derivative $f^{\prime }\left(x\right)$ is different.