Chapter 11.3, Problem 76E

To determine

To match: Match the given function with the graph of its derivative.

Answer to Problem 76E

Derivative of the given function	The graph of the derivative
$(a).f`\left(x\right)=\frac{d}{dx}\left(\sqrt{x}\right)=\frac{1}{2\sqrt{x}}$
$(b).f`\left(x\right)=\frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2}$
$(c).f`\left(x\right)=\frac{d}{dx}\left(x\right)=1$
$(d).f`\left(x\right)=\frac{d}{dx}\left(x^3\right)=3x^2$

Explanation of Solution

Given information: The functions are

  $(a).f\left(x\right)=\sqrt{x},(b).f\left(x\right)=\frac{1}{x},(c).f\left(x\right)=x,(d).f\left(x\right)=x^3$

Formula used: $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Calculation:

Calculate the derivative and draw the graph of (a):

  $\begin{array}{l}\therefore f\left(x\right)=\sqrt{x}\\ =\frac{d}{dx}\left(\sqrt{x}\right)\\ =\frac{d}{dx}\left(x^{\frac{1}{2}}\right)\\ =\frac{1}{2}{\left(x\right)}^{1-\frac{1}{2}}\\ =\frac{1}{2\sqrt{x}}\end{array}$

  

Calculate the derivative and draw the graph of (b):

  $\begin{array}{l}\therefore f\left(x\right)=\frac{1}{x}\\ =\frac{d}{dx}\left(x^{-1}\right)\end{array}$

  $=\frac{d}{dx}\left(-x^{-1-1}\right)$

  $\begin{array}{l}=-{\left(x\right)}^{-2}\\ =-\frac{1}{x^2}\end{array}$

  

Calculate the derivative and draw the graph of (c):

  $\begin{array}{l}\therefore f\left(x\right)=x\\ =\frac{d}{dx}\left(x^1\right)\\ =\left(x^{-1+1}\right)\\ =x^0\\ =1\end{array}$

  

Calculate the derivative and draw the graph of (d):

  $\begin{array}{l}\therefore f\left(x\right)=x^3\\ =\frac{d}{dx}\left(x^{{}^3}\right)\\ =\left(3x^{3-1}\right)\\ =2x^2\end{array}$