Question-40
differentiability
Consider the function \(\displaystyle f( x) =3^{x}\). Find \(\displaystyle f^{\prime}( 1)\).
Hint
Take \(\log\) on both sides and differentiate.
Answer
Solution
Let \(\displaystyle y=3^{x}\), then:
\[ \begin{equation*} \begin{aligned} \ln y & =x\ln 3\\ \frac{1}{y}\frac{dy}{dx} & =\ln 3\\ \Longrightarrow \frac{dy}{dx} & =y\ln 3 \end{aligned} \end{equation*} \]
Expressing this in terms of \(\displaystyle f\) and \(\displaystyle x\):
\[ \begin{equation*} f^{\prime }( x) =3^{x}\ln 3 \end{equation*} \]
Therefore \(\displaystyle f^{\prime }( 1) =3\ln 3\).