Question

How are derivatives related to area?

Answer 1

See below.

Explanation:

Derivatives aren't directly related or associated with area.

Derivatives describe the `slope` of a tangent line in relation to another line, at a given point.

However, derivatives can relate to the rate of change in an objects area. This is usually figured through [implicit differentiation.](https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-advanced/ab-implicit-diff/v/implicit-differentiation-1)

This finds the rate at which an area is increasing/decreasing, by finding the derivative of the appropriate area formula in respect to time.

For explanation sake, here is an example:

Let's say we need to find the rate in which the `area` of a circle is increasing, when the rate of the `radius` increasing is `2`m/s with the `radius` being at `4` meters.

We know that the `area` of a circle is:

`A=πr^2`

Using the chain rule , we can find the derivative of the equation. However, since we are finding the derivative in respect to time, we need to take on a `(d(square))/dt` (`square` being the variable in which you are finding the derivative of.

So, here is what it will look like after we find the derivative (See above link if you have questions):

`(dA)/dt=2πr(dr)/dt`

From here we can simply plug in our values:

`(dA)/dt=2π(4)(2)`

`(dA)/dt=16π m^2 / sec`