Question

What is the derivative of ${{10}^{x}}$ ?

Answer 1

Differentiation is a method of finding the derivative of a function. Differentiation is a process where we find the instantaneous rate of change in function based on one of it’s variables. The most common example is the rate of change of displacement with respect to time is called velocity. The opposite of finding derivatives is called integration. If $x$ is a variable and $y$is another variable, then the rate of change of $y$ with respect to $x$ is denoted as $\dfrac{dy}{dx}$ .

Complete step by step solution:
For example we have a function. Let it be $f\left( x \right)={{a}^{x}}$ .
The derivative of this function with respect to $x$ is the following :
$\begin{aligned} & \Rightarrow y={{a}^{x}} \\\ & \Rightarrow \dfrac{dy}{dx}={{a}^{x}}\log x \\\ \end{aligned}$
This is a formula for the derivative of the function of the type ${{a}^{x}}$ where $a$ is a constant.
Just like how we have a formula for the derivative of ${{x}^{n}}$.
The derivative of ${{x}^{n}}$ is the following :
$\Rightarrow \dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ .
These formulae are derived from the first rule of differentiation.
Let us compare.
Upon comparing the general form of the formulae with the function given in the question, we get our $a$ as $10$ and our variable in both is $x$.
So let us apply the formula and find out the derivative.
Upon doing so, we get the following :
$\begin{aligned} & \Rightarrow y={{10}^{x}} \\\ & \Rightarrow \dfrac{dy}{dx}={{10}^{x}}\log x \\\ \end{aligned}$

$\therefore$ The derivative of ${{10}^{x}}$ is ${{10}^{x}}\log x$.

Note: We should be aware of all the different rules of differentiation. We should also know all the formulae of differentiation so as to be able to complete a question quickly in the exam. The above question can also be done using the first rule of differentiation but it is quite lengthy and time taking. Hence, we just use the formula because it is already derived. We should also know the expansions of some function so as to find the derivative of them.