Question

What is the derivative of ${{5}^{x}}$ ?

Answer 1

To obtain the derivative of ${{5}^{x}}$ we will use logarithm function. Firstly we will let $y={{5}^{x}}$ and then take a logarithm function on both sides of the equation. Next we will use the logarithm property and simplify our equation. Finally we will differentiate the equation with respect to $x$ using implicit differentiation and get the desired answer.

Complete step-by-step answer:
To find the derivative of ${{5}^{x}}$ we will let,
$y={{5}^{x}}$…..$\left( 1 \right)$
Taking $\log$ both sides we get,
$\log \left( y \right)=\log \left( {{5}^{x}} \right)$
Using logarithm property on right hand side which is given as:
$\log \left( {{a}^{b}} \right)=b\log \left( a \right)$
Where $a,b$ can be any constant or variable
We get,
$\log \left( y \right)=x\log \left( 5 \right)$…..$\left( 2 \right)$
Differentiating equation (2) with respect to $x$ using implicit differentiation we get,
$\begin{aligned} & \log \left( y \right)=x\log \left( 5 \right) \\\ & \Rightarrow \dfrac{1}{y}\times \dfrac{dy}{dx}=\log \left( 5 \right)\times \dfrac{d\left( x \right)}{dx} \\\ & \Rightarrow \dfrac{1}{y}\times y=\log \left( 5 \right)\times 1 \\\ & \therefore {y}'=y\log \left( 5 \right) \\\ \end{aligned}$
Where, Primes $\left( ' \right)$ denote the differentiation with respect to $x$
Replacing $y$ value from equation (1) in above equation we get,
$\begin{aligned} & \Rightarrow {{y}^{'}}={{5}^{x}}\log \left( 5 \right) \\\ & \therefore {{\left( {{5}^{x}} \right)}^{'}}={{5}^{x}}\log \left( 5 \right) \\\ \end{aligned}$
Hence, the derivative of ${{5}^{x}}$ is ${{5}^{x}}\log \left( 5 \right)$

Note: An exponential function is expressed as $f\left( x \right)={{a}^{x}}$ where $a$ is a positive real number and $x$ is an argument which is present as an exponent. The growth rate of such a function is directly proportional to the value of the function. Implicit differentiation is done when we can’t find the derivative of $y$ in $x$ term, that is $x$ doesn’t lead to $y$ directly. We can use another method to find the derivative of the exponential function by using the direct formula of it. To find the derivative of ${{5}^{x}}$ we can also use the derivative formula for exponential function as,
The formula is given below:
$\dfrac{d}{dx}{{a}^{x}}={{a}^{x}}{{\log }_{e}}a$….$\left( 3 \right)$ For $a\in R$
On comparing the above equation by ${{5}^{x}}$ we get,
$a=5$
On substituting the above value in equation (3) we get,
$\Rightarrow \dfrac{d}{dx}{{5}^{x}}={{5}^{x}}{{\log }_{e}}5$
Hence, the derivative of ${{5}^{x}}$ is${{5}^{x}}{{\log }_{e}}5$.