Question

What is the derivative of \[\pi .{r^2}\]?

Answer 1

Hint:We need to find the derivative of \[\pi .{r^2}\]. We will be finding the derivative of this expression with respect to \[r\] using the formulas for differentiation. We will be using power rule and property involving constant term, which are as follows:

Formula used:
POWER RULE - \[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\]
PROPERTY INVOLVING CONSTANT - \[\dfrac{d}{{dx}}\left( {c.f\left( x \right)} \right) = c.\left( {\dfrac{d}{{dx}}\left( {f\left( x \right)} \right)} \right)\], where \[c\] is a constant term.

Complete step by step answer:
We need to find the derivative of \[\pi .{r^2}\] with respect to \[r\].
Let \[y = \pi .{r^2} - - - - - - - (1)\]
So, we need to find \[\dfrac{{dy}}{{dr}}\]
Differentiating (1) with respect to \[r\], we get
\[ \Rightarrow \dfrac{{dy}}{{dr}} = \dfrac{d}{{dr}}\left( {\pi .{r^2}} \right)\]

Here we see that \[\pi \] is a constant term. Hence using the Property
\[\dfrac{d}{{dx}}\left( {c.f\left( x \right)} \right) = c.\left( {\dfrac{d}{{dx}}\left( {f\left( x \right)} \right)} \right)\], where \[c\] is a constant term, we get
\[ \Rightarrow \dfrac{{dy}}{{dr}} = \pi .\left( {\dfrac{d}{{dr}}{r^2}} \right)\]

Now using the Property \[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\], we get
\[ \Rightarrow \dfrac{{dy}}{{dr}} = \pi .\left( {2{r^{2 - 1}}} \right)\]
\[ \Rightarrow \dfrac{{dy}}{{dr}} = \pi .\left( {2{r^1}} \right)\]
\[ \Rightarrow \dfrac{{dy}}{{dr}} = \pi .\left( {2r} \right)\]
As multiplication is commutative we can write the above expression as
\[ \therefore \dfrac{{dy}}{{dr}} = 2\pi r\]

Hence derivative of \[\pi .{r^2}\] with respect to \[r\] is \[2\pi r\].

Note:We can also solve the given problem using First Principle of Differentiation.According to First Principle of differentiation, the derivative of a function \[f\left( x \right)\] can be evaluated by calculating the limit \[f'\left( r \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {r + h} \right) - f\left( r \right)}}{h}\], where \[f'\left( r \right)\] is the first derivative of the function \[f\left( r \right)\] with respect to \[r\].