Question

What is the derivative of $\dfrac{\pi }{x}$ ?

Answer 1

To solve this question we need to know the concept of differentiation and its formula. The formula used to evaluate the above function is $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ . The function consists of the constant variable $\pi$.

Complete step by step solution:
The question asks us to find the derivative of $\dfrac{\pi }{x}$ which means we need to differentiate the function given to us which is $\dfrac{\pi }{x}$ . Here $\pi$ is a constant with Value 3.14 so while of the differentiation of $\dfrac{\pi }{x}$ will be used as the constant so we will be just differentiating of $\dfrac{1}{x}$ with respect to $x$.
On differentiating the function $\dfrac{\pi }{x}$ we get:
$\Rightarrow \dfrac{d\left( \dfrac{\pi }{x} \right)}{dx}$
In the above function $\pi$will be taken out of the function as given below:
$\Rightarrow \pi \dfrac{d\left( \dfrac{1}{x} \right)}{dx}$
The above fraction that need to be differentiated will be reciprocal and the power would as a result change to $-1$, this could be written as:
$\Rightarrow \pi \dfrac{d\left( {{x}^{-1}} \right)}{dx}$
To solve the above expression we will use the formula, $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ where value of “n” is $-1$. On applying the above formula to the function we get:
$\Rightarrow \pi \left( -1 \right){{x}^{-1-1}}$
On further calculation the above expression becomes:
$\Rightarrow -\pi {{x}^{-2}}$
The above answer could be changed to the term with the positive power of $x$ by doing its reciprocal.
$\Rightarrow -\dfrac{\pi }{{{x}^{2}}}$

$\therefore$ The derivative of $\dfrac{\pi }{x}$ is $\dfrac{-\pi }{{{x}^{2}}}$

Note: The above question $\pi$ is constant in the function given to us. Do get confused with the value of $\pi$ . We need to remember the formula for the differentiation of the function given above. Do remember that the reciprocal of a value changes the sign of the power. For example if a function or number ${{a}^{-1}}$ is reciprocal the power changes to positive sign like ${{\left( \dfrac{1}{a} \right)}^{1}}$.