#### Provide solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Very short answers question 6

The answer of the given question will be $\frac{1}{9}$.

Given:

Let $g(x)$ be the inverse of an invertible function) $f(x)$ which is derivable at $x=3$. If $f (3) =9$ and $f^{'} (3) =9$ find the value of $g^{'}(9)$.

Hint:

\begin{aligned} &g(x) \stackrel{\text { invertible }}{\longrightarrow} f(x) \\\\ &g \circ f(x) \stackrel{\text { invertible }}{\longrightarrow} I(x) \end{aligned}

Solution:

Differentiating both sides we get,

\begin{aligned} &g \circ f^{\prime}(x)=1 \\\\ &\Rightarrow \frac{d}{d x} g[f(x)]=1 \\\\ &\Rightarrow g^{\prime}[f(x)]=f^{\prime}(x)=1 \end{aligned}

Now $x=3$

\begin{aligned} &g^{\prime}[f(3)]=\frac{1}{f^{\prime}(3)}[\therefore f(3)=9] \\\\ &\Rightarrow g^{\prime}(9)=\frac{1}{9}\left[f^{\prime}(3)=9\right] \end{aligned}

So the answer will be  $\frac{1}{9}$