Constant Rule for finding Derivatives

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.

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:

Since $f(x)=c$ and $f(x+h)=c$ (the function always returns the same value), we can substitute these into the limit definition:

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 $f(x)=5x^3+2x-7$.

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:

Using the Power Rule for the first two terms and the Constant Rule for the last term, we get:

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