What if we want to know the exact gradient at some point on a curve?

We can actually do that by considering what happens when the secant actually becomes the tangent. That is, we consider what happens to

as $h$ tends to $0.$

This can be written using limit notation as

After finding a formula for $\frac{dy}{dx},$ we can then use it to calculate the gradient at any desired point on the curve! This is much more convenient.

The whole process of finding $\frac{dy}{dx}$ in this way is called differentiation from first principles. Let's see an example of this.

A curve is given by

Use differentiation from first principles to find a formula for

and hence find the gradient at the point on the curve where $x=3.$