Question

What is the derivative of ${{7}^{x}}$?

Answer 1

Hint: Assume the given function as $y=f\left( x \right)$. Take log to the base e both the sides and use the property of logarithm given as $\ln {{a}^{m}}=m\ln a$ to simplify. Now, differentiate both the sides of the assumed function y and use the chain rule of derivative to find the derivative of L.H.S. Use the formula for the derivative of the natural log function given as $\dfrac{d\ln x}{dx}=\dfrac{1}{x}$ for the simplification. Finally, substitute back the assumed value of y to get the value of $\dfrac{dy}{dx}$.

Complete step by step answer:
Here we have been provided with the function ${{7}^{x}}$ and we are asked to differentiate it. Let us assume this function as y so we have,
$\Rightarrow y={{7}^{x}}$
Now, we need to find the value of $\dfrac{dy}{dx}$. Taking natural log, i.e. log to the base e, on both the sides we get,
$\Rightarrow \ln y=\ln \left( {{7}^{x}} \right)$
Using the property of log given as $\ln {{a}^{m}}=m\ln a$ we get,
$\Rightarrow \ln y=x\ln 7$
Differentiating both the sides with respect to x we get,
$\Rightarrow \dfrac{d\ln y}{dx}=\dfrac{d\left( x\ln 7 \right)}{dx}$
Here $\ln 7$ is a constant so it can be taken out of the derivative, so we get,
\[\begin{align}
  & \Rightarrow \dfrac{d\ln y}{dx}=\ln 7\times \dfrac{d\left( x \right)}{dx} \\
 & \Rightarrow \dfrac{d\ln y}{dx}=\ln 7\times 1 \\
 & \Rightarrow \dfrac{d\ln y}{dx}=\ln 7 \\
\end{align}\]
Using the chain rule of derivative in the L.H.S where we will differentiate $\ln y$ with respect to y and then its product will be taken with the derivative of y with respect to x, so we get,
\[\Rightarrow \dfrac{d\ln y}{dy}\times \dfrac{dy}{dx}=\ln 7\]
Using the formula of the derivative of natural log function given as $\dfrac{d\ln x}{dx}=\dfrac{1}{x}$ we get,
$\begin{align}
  & \Rightarrow \dfrac{1}{y}\times \dfrac{dy}{dx}=\ln 7 \\
 & \Rightarrow \dfrac{dy}{dx}=y\ln 7 \\
\end{align}$
Substituting back the assumed function y we get,
$\therefore \dfrac{dy}{dx}={{7}^{x}}\ln 7$

Hence the derivative of the given function is ${{7}^{x}}\ln 7$.

Note: You can also remember the direct formula for the derivative of the function of the form ${{a}^{x}}$ called the exponential function. The formula is given as $\dfrac{d\left( {{a}^{x}} \right)}{dx}={{a}^{x}}\ln a$ where ‘a’ is any constant. If a = e then the derivative formula becomes $\dfrac{d\left( {{e}^{x}} \right)}{dx}={{e}^{x}}$ because $\ln e=1$. Note that you must not take log with base 10 or any other base because we don’t have a direct formula for that. Even if we do so then we need to apply the base change rule of log that will only increase the steps of the solution.