Common Derivatives > Pi Derivative

**The derivative of π with respect to x is 0.**

π is a constant (equal to about 3.14), and the derivative of a constant is zero. Therefore π^{2}, π/2, π/6, π/10 or any other π/constant variants all have derivatives of zero.

**This only works if you have a constant**. It doesn’t work if you have any variable in your expression. For example, the rule doesn’t apply to π(x), π^{2}(x) or πsin(x).

There are some other interesting relationships with derivatives involving π, especially the one involving the derivative of the area of a circle.

## Pi Derivative in Other Functions

Examples:

- π
^{2}(x) - x
^{π} - π
^{x}

**Example #1 :** What is the derivative with respect to x of

f(x) = π^{2}(x)?

Step 1: Place the constant in front:

= π^{2} ^{d}/_{dx}(x).

Step 2: Use the common pi derivative ^{d}/_{dx} = 1:

= π^{2} · 1.

Step 3: Simplify:

= π^{2}.

**Example #2 :** What is the derivative with respect to x of

f(x) = x^{π}?

Step 1: Use the power rule ^{d}/_{dx} (x^{a} = a · x^{a – 1}):

We have a = π so:

= πx^{π-1}

*That’s it!*

**Example #3 :** What is the derivative with respect to x of f(x) = π^{x}?

Step 1: Use the common derivative rule for exponential functions ^{d}/_{dx}(a^{x} = a^{x}ln(a)):

Plugging our function into the formula, we have a = π, so:

= π^{x}ln(π).

These examples demonstrate that we can use common rules to simplify functions involving a pi derivative.

## Derivative of a Circle’s area πr^{2}

**The derivative of a circle’s area (πr ^{2}) is it’s circumference (2*πr).**

This relationship also holds for a semicircle, and it can be extended to a sphere: the derivative of the volume function of a sphere equals its surface area. This interesting derivative relationship does not hold for all shapes though, such as squares or rectangles [1].

The math behind this fact is used in the cylindrical shell method for finding volumes of shapes. The logic is as follows [2]: a small change in the radius of the sphere produces a small change in the sphere’s volume, which is equal to the volume of a spherical shell of radius R and thickness δR. This shell’s volume is approximately:

V = Surface area of sphere * δ R.

The derivative here is the change in ball volume / δR, which is just the surface area of the sphere.

## Pi Derivative: References

[1] Marichal, J. & Dorff, M. (2007). Derivative Relationships between volume and surface area of compact regions in R^{d}. Rocky Mountain Journal of Mathematics, vol 37 No 2. Retrieved May 3, 2021 from: https://math.byu.edu/~mdorff/docs/SomeRelations2007.pdf

[2] Su, Francis E., et al. “Surface Area of a Sphere.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.