Question

What is the derivative of $3^{x}$ ?

Answer 1

In this question , we need to find the derivative of $3^{x}$. Mathematically, derivative is nothing but a rate of change of function with respect to an independent variable given in the function. Differentiation is used to calculate the derivatives. In two ways we can find the derivative of $3^{x}$. We can find the derivative by taking logs on both sides. Also there is a direct formula to find the derivative of $3^{x}$ .
Logarithmic rule used :
${log }\ m^{n} = nlog\ m$
Derivative formula used :
1.$\dfrac{d}{{dx}}\ (logx) =\dfrac{1}{x}$
2. $\dfrac{d}{{dx}}\ (x) =1$

Complete step-by-step solution:
Given ,
$3^{x}$
Let us consider , $y = 3^{x}$
On taking log on both sides,
We get,
$log\ y = log\left( 3^{x} \right)$
We know that ${log }\ m^{n} = nlog\ m$
By using that logarithmic rule,
We get,
$log\ y = x\ log\ 3$
On differentiating $y$ with respect to $x$,
We get ,
$\dfrac{dy}{{dx}}\left({log \ y} \right) = \dfrac{{dy}}{{dx}}\left( x\ log\ 3 \right)$
⇒ $\dfrac{1}{y}\dfrac{{dy}}{{dx}} = 1 \times \left( log\ 3 \right)$
By cross multiplying,
We get,
$\dfrac{dy}{{dx}} = y(log\ 3)$
We have already considered $y = 3^{x}$,
By substituting the value of $y$,
We get,
$\dfrac{dy}{{dx}} = 3^{x}(log\ 3)$
Thus we get the derivative of $3^{x}\$ is $3^{x}(log\ 3)$
Final answer :
The derivative of $3^{x}$ is $3^{x}\left( log\ 3 \right)$

Note: Derivative helps in solving the problems in calculus and in differential equations. The derivative of $y$ with respect to $x$ is represented as $\dfrac{{dy}}{{dx}}$. There are two types of derivative namely first order derivative and second order derivative. A simple example for a derivative is the derivative of $x^{3}$ is $3x$ . Derivative is also applicable in trigonometric functions.
Alternative solution :
We can also find the derivative of $3^{x}$ by using a derivative formula.
Formula used :
$\dfrac{d}{{dx}}\left( a^{x} \right) = \ a^{x}\log a$
Given,
$3^{x}$
Here $a = 3$
By substituting the value of $a$ in the formula,
We get,
$\dfrac{d}{{dx}}\left( 3^{x} \right) = \ 3^{x}\log 3$
Thus we get the derivative of $3^{x}$ is $3^{x}\left( log\ 3 \right)$