The Constant Rule is a differentiation rule used to find the derivative of a **constant function.** A constant function is a function that **always returns the same value,** regardless of the input. In other words, the graph of a constant function is a horizontal line. The Constant Rule states that the derivative of a constant function is **always zero.**

## Constant Rule Formula and Formal Definition

The Constant Rule formula is as follows:

$$\frac{\text{d}}{\text{d}x}(c)=0$$Let $c$ be a constant. If $f(x)=c$, then the derivative of $f(x)$ with respect to $x$ is given by:

$${f}^{\prime}(x)=\frac{\text{d}}{\text{d}x}(c)=0$$## Deriving the Constant Rule

To understand why the Constant Rule works, let’s derive it using the limit definition of the derivative. Given a constant function $f(x)=c$, we have:

$${f}^{\prime}(x)={lim}_{h\to 0}\frac{f(x+h)-f(x)}{h}$$Since $f(x)=c$ and $f(x+h)=c$ (the function always returns the same value), we can substitute these into the limit definition:

$${f}^{\prime}(x)={lim}_{h\to 0}\frac{c-c}{h}={lim}_{h\to 0}\frac{0}{h}={lim}_{h\to 0}0=0$$So we have shown that the derivative of a constant function is always zero.

## Significance of the Constant Rule

The Constant Rule is a great way to understand the behaviour of more complex functions. It’s really useful for figuring out whether a polynomial or rational function is constant or power-based (e.g., $g(x)={x}^{n}$, where $n$ is a positive integer).

By understanding that the derivative of a constant term is always zero, we can make the process of finding derivatives for these more complex functions much easier. Let’s take a look at an example. When we differentiate a polynomial, the Constant Rule helps us to focus on the non-constant terms. This is because the derivative of the constant term will always be zero.

### Example: Applying the Constant Rule

Let’s consider the function $f(x)=5{x}^{3}+2x-7$.

To find the derivative of this function, we can apply the Constant Rule to the constant term and the Power Rule to the non-constant terms:

$${f}^{\prime}(x)=\frac{\text{d}}{\text{d}x}(5{x}^{3})+\frac{\text{d}}{\text{d}x}(2x)-\frac{\text{d}}{\text{d}x}(7)$$Using the Power Rule for the first two terms and the Constant Rule for the last term, we get:

$${f}^{\prime}(x)=15{x}^{2}+2-0=15{x}^{2}+2$$Thus, the derivative of $f(x)=5{x}^{3}+2x-7$ is ${f}^{\prime}(x)=15{x}^{2}+2$.