Question

A function $f\left(x\right)$ is defined for all $x>0$and satisfies$f\left(x^2\right)=x^3$ for all $x>0$. Then $f’\left(4\right)$ is equal to

• A. $1$
• B. $2$
• C. $3$
• D. None of these

Answer 1

The correct option is C

$3$

Explanation for the correct answer:

Step 1: Finding the derivative

The function is $f\left(x^2\right)=x^3,x>0$

Differentiate both sides with respect to x,

$$\begin{gathered} \Rightarrow \frac{d\left[f\left(x^2\right)\right]}{dx}=\frac{d\left[x^3\right]}{dx} \\ \Rightarrow f\text{'}\left(x^2\right)\left(2x\right)=3x^2\left[\because \frac{d\left[x^2\right]}{dx}=2x\right] \\ \Rightarrow f\text{'}\left(x^2\right)=\frac{3x^2}{2x} \\ \Rightarrow f\text{'}\left(x^2\right)=\frac{3}{2}x \end{gathered}$$

Step 2: Finding the value of $x$

$$\begin{gathered} \Rightarrow f\text{'}\left(x^2\right)=f\text{'}\left(4\right) \\ \Rightarrow x^2=4 \\ \Rightarrow x=\sqrt{4} \\ \Rightarrow x=2 \end{gathered}$$

Step 3: Finding the value of $f\text{'}\left(4\right)$

Substitute the value of x, we get

$$\begin{gathered} f\text{'}\left(4\right)=\frac{3}{2}\left(2\right) \\ =3 \end{gathered}$$

Hence, option (C) is the correct answer.