Question

How do you differentiate ${x^\pi }$ ?

Answer 1

We are given a function that involves x raised to some power so the given function is in terms of x.
We must know what differentiation actually is for differentiating the given function. When we divide a whole quantity into very small ones, the process is known as differentiation, in the given question we have to differentiate ${x^\pi }$ with respect to x. So, we let ${x^\pi }$ to be represented by some unknown quantity “y”. This way x becomes the independent variable and y becomes the dependent variable.
When we have to find the instantaneous rate of change of a quantity we use differentiation, it is represented as $\dfrac{{dy}}{{dx}}$ , in the expression $\dfrac{{dy}}{{dx}}$ , a very small change in quantity is represented by $dy$ and the small change in the quantity with respect to which the given quantity is changing.
Is represented by $dx$ .

Complete step by step answer:
We have to differentiate ${x^\pi }$
Let $y = {x^\pi }$ , so we have to differentiate the function y.
We know that the derivative of ${x^n}$ is $n{x^{n - 1}}$ so,

$$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}({x^\pi }) \\\ \Rightarrow \dfrac{{dy}}{{dx}} = \pi {x^{\pi - 1}} \\\$$

We know that $\pi \approx 3.14$
Hence the derivative ${x^\pi }$ of is nearly equal to $3.14{x^{2.14}}$ .

Note: Letting $y = {x^\pi }$ makes the representation of differentiation easier and convenient. We must note that x is raised to power $\pi$ , here $\pi$ is a constant. So the students should not get confused while differentiating the given function. And while solving similar questions, we must rearrange the equation first, so that the variable with respect to which the function is differentiated is present on one side and the variable whose derivative we have to find is present on the other side.