Derivative : Product Rule

Introduction

In calculus, sometimes we need to find the derivative of a function that is the product of two other functions.
For example, what if we want the derivative of:

$$f(x)=(x+5)(2x+7)$$

We can’t just take the derivative of each part separately — that would give the wrong answer!
This is where the Product Rule comes in.

Why We Need the Product Rule

If $$f(x)=u(x)⋅v(x)$$, where both u and v are differentiable functions ,we can’t simply say:

$$f^\prime(x)=u^\prime(x)v^\prime(x)$$

That’s incorrect.
Instead, we need to apply the Product Rule, which tells us how to correctly differentiate products of two functions.

The Product Rule Formula

If: $$f(x)=u(x)⋅v(x)$$

then: $$f^\prime(x)=u^\prime(x)v(x)+u(x)v^\prime(x)$$

and if we have function which is products of three functions

$$g(x)=u(x)⋅v(x)⋅z(x)$$ then

$$g^\prime(x)=u^\prime(x)v(x)z(x)+u(x)v^\prime(x)z(x)+u(x)v(x)z^\prime(x)$$