Question

Find the derivative of function as $f'(x)$ for $f(x)={{5}^{x}}$?

Answer 1

To solve the given problem, we should know how to evaluate the derivative of the following functions. The derivative of $\ln x$ is $\dfrac{1}{x}$. We should also know that constants can be taken out of the differentiation if they are multiplied with a function, algebraically we can express it as $\dfrac{d\left( af(x) \right)}{dx}=a\dfrac{d\left( f(x) \right)}{dx}$.

Complete step by step solution:
we are given the function $f(x)={{5}^{x}}$, we are asked to find its derivative $f'(x)$. To make things simple, let’s put $y=f(x)$. Now, we need to evaluate $\dfrac{dy}{dx}$.
$y={{5}^{x}}$
Taking logarithm of both sides of above equation, we get
$\Rightarrow \ln y=\ln {{5}^{x}}$
We know the property of logarithm that $\ln {{a}^{n}}=n\ln a$. Using this we can simplify the above expression as
$\Rightarrow \ln y=x\ln 5$
Taking derivative of both sides, we get
$\Rightarrow \dfrac{d\left( \ln y \right)}{dx}=\dfrac{d\left( x\ln 5 \right)}{dx}$
As $\ln 5$ is a constant, we can take it out of the differentiation simplifying above expression as
$\Rightarrow \dfrac{d\left( \ln y \right)}{dx}=\ln 5\dfrac{d\left( x \right)}{dx}$
We know that the derivative of the function $\ln x$ with respect to x is $\dfrac{1}{x}$, and the derivative of x with respect to x is 1. Using these, we can evaluate the above differentiation as
$\Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=\ln 5$
Multiplying $y$ to both sides of above equation, we get
$\Rightarrow \dfrac{dy}{dx}=y\ln 5$
Substituting the given function for $y$, we get
$\Rightarrow \dfrac{dy}{dx}={{5}^{x}}\ln 5$
Thus, the derivative of a given function is ${{5}^{x}}\ln 5$.

Note: We can use the given problem to make a differentiation property for these types of functions. For a given function of the form $y={{a}^{x}}$. The derivative can be evaluated as
$\Rightarrow \dfrac{dy}{dx}={{a}^{x}}\ln a$
As the domain of logarithmic functions is $(0,\infty )$. The $a$ must also belong to this range.