Question

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

Answer 1

Hint: 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.