Chapter 4.4, Problem 2QQ

To determine

To calculate: The derivative of given function.

Answer to Problem 2QQ

The option (B) is correct, which is $\frac{1}{4}$ .

Explanation of Solution

Given information: The equation of f(x) in terms of x is given as,

  $\begin{array}{l}f\left(x\right)=x^3+x\\ g\left(x\right)=f^{-1}\left(x\right)\\ g\left(2\right)=1\end{array}$

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

Calculation: Using the derivation,

  $\begin{array}{l}f(x)=x^3+x\\ \text{Take}{y}_1=f\left(x\right){\text{and y}}_2=g\left(x\right)\\ y_2=f^{-1}\left(x\right)\\ x=y_2{}^3+y_2\\ \text{Differentiate both sides,}\\ \frac{dx}{dx}=\frac{d\left(y_2{}^3+y_2\right)}{dx}\\ 1=\left(3y_2{}^2+1\right)\frac{d{y}_2}{dx}\\ \frac{d{y}_2}{dx}=\frac{1}{3y_2{}^2+1}\\ \text{Now put }x=2,\\ g\text{'}\left(2\right)=\frac{d{y}_2}{dx}\\ g\text{'}\left(2\right)=\frac{1}{3{\left(g\left(x\right)\right)}^2+1}\\ g\text{'}\left(2\right)=\frac{1}{3{\left(g\left(2\right)\right)}^2+1}\\ g\text{'}\left(2\right)=\frac{1}{3{\left(1\right)}^2+1}\\ g\text{'}\left(2\right)=\frac{1}{4}\end{array}$

So the derivative is $\frac{1}{4}$ .