Question

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

Answer 1

Hint: 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.