Derivative of a Function — What Is It?

Author(s): Sujeeth Kumaravel

Originally published on Towards AI.

What is a tangent to a function f(x) at point x? It is the line that touches the function only at point x. It doesn’t intersect the function at any other point x2, which is different from x.

What is the derivative of a function f(x) at point x? It is the slope of the tangent to the function at that point.

Consider the function:

The graph of this function is:

For any x < 0, the tangent to the function at x would look like:

This tangent has a negative slope. So, the derivative of f(x) at x < 0 is negative.

For any x > 0, the tangent to the function at x would look like:

This tangent has a positive slope. So, the derivative of f(x) at x > 0 is positive.

At x = 0, the tangent to the function coincides with the x-axis as shown below:

This tangent has a slope of 0. So, the derivative of f(x) at x = 0 is zero.

Note that at x = 0, the tangent will not be like the following because, the line intersects the function at 2 points:

The derivative of a function f(x) is mathematically defined as the following limit:

Here h is an infinitesimally small number.

As h approaches 0 from the negative direction (for negative values of h) the following is the value of the limit:

As h approaches 0 from the positive direction (for positive values of h) the following is the value of the limit:

The derivative is denoted as:

These two limits must be equal. That limit value is the derivative.

Here df(x) at x = a is:

Here dx is:

Note that the example taken above is a function that is continuous and smooth at all x. Derivatives of a function exist only at points where it is continuous and smooth and where vertical tangents don’t occur (at points where vertical tangents occur, slope of the tangent is infinity). A derivative is also called differential.

There are points where a function will not have a derivative. The function is not differentiable at such points. We will see such points in a future post.

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Published via Towards AI