Question

How do you find the derivative of exponential function $y = {e^9}\ln x$?

Answer 1

In this the function y is given we have to find the derivative of the exponential function. To solve this, we have to differentiate the given function with respect to x using the standard differential formula of log and exponential function. On further simplification we get the required solution.

Complete step by step solution:
The differentiation of a function is defined as the derivative or rate of change of a function. The function is said to be differentiable if the limit exists.
Differentiation can be defined as a derivative of a function with respect to an independent variable.
Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by: $\dfrac{{dy}}{{dx}}$
Consider the given exponential function
$\Rightarrow \,\,y = {e^9}\ln x$--------(1)
We have to find derivative $\dfrac{{dy}}{{dx}} = ?$
The exponential term in the given function i.e., ${e^9}$ is a constant term.
Differentiate equation (1), with respect to x, then
$\Rightarrow \,\,\dfrac{{dy}}{{dx}} = {e^9}\dfrac{d}{{dx}}\left( {\ln x} \right)$---------(2)
The differentiation of log x is $\dfrac{d}{{dx}}\left( {\ln x} \right) = \dfrac{1}{x}$
Then, equation (2) becomes
$\Rightarrow \,\,\dfrac{{dy}}{{dx}} = {e^9}\dfrac{1}{x}$
Or
$\Rightarrow \,\,\dfrac{{dy}}{{dx}} = \dfrac{{{e^9}}}{x}$

Hence, the differentiated term of the given exponential function is $\dfrac{{dy}}{{dx}} = \dfrac{{{e^9}}}{x}$.

Note: The differentiation is the rate of change of a function at a point. We must know about the chain rule of derivatives. The function can be written as a composite of two functions, if the function can be written as a composite of two functions then we can apply the chain rule of derivative. By using log to the terms we can differentiate the function in an easy manner.