[Solved] What is the Derivative of pi/4?

We know that the value of the number $\pi$ is approximately equal to 3.1416 (up to $4$ decimal places). Also, the number $\pi$ is irrational.

 

Note that the value of $\pi$ is determined by the area of a unit circle (that is, a circle of radius 1). As the area of a unit circle is fixed, so we conclude that $\pi$ is a fixed number.

 

$\Rightarrow \pi$ is a constant.

 

$\Rightarrow \dfrac{\pi}{4}$ is a constant.

 

So $\dfrac{\pi}{4}$ does not change with respect to any variable.

 

$\therefore \dfrac{d}{dx}(\dfrac{\pi}{4})=0$ by the rule Derivative of a constant is 0.

 

Thus, the derivative of $\dfrac{\pi}{4}$ is equal to $0$.

 

 

Derivative of pi/4 by First Principle

Let $f(x)=\dfrac{\pi}{4}$. As both $\pi$ and $4$ are constants, the quotient $\dfrac{\pi}{4}$ is independent of $x$. Thus, we have

 

$f(x+h)=\dfrac{\pi}{4}$ for any values of $x$ and $h$.

 

Now, by the first principle, the derivative of $f(x)$ is equal to

 

$\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$

 

In this formula, we put $f(x)=\dfrac{\pi}{4}$.

 

Hence $\dfrac{d}{dx}(\dfrac{\pi}{4})$ $=\lim\limits_{h \to 0} \dfrac{\dfrac{\pi}{4} -\dfrac{\pi}{4}}{h}$

 

$=\lim\limits_{h \to 0} \dfrac{0}{h}$

 

$=\lim\limits_{h \to 0} 0$

 

$=0$.

 

Thus, the derivative of  $\dfrac{\pi}{4}$ from the first principle, that is, by the limit definition is equal to $0$.