As we discussed in Chapter 17 (Derivatives of Common Functions and The Chain Rule), it is possible to differentiate functions other than polynomials. However, what should we do when we are given a function that is two different functions being multiplied together? Or, how are we to differentiate a function that is two different functions where one function is being divided by the other function? Fortunately, rules have been established where we can differentiate the product of two functions. Likewise, we are able to differentiate a function that is the quotient of two other functions. These are referred to as The Product Rule and The Quotient Rule.

The Product Rule: Given a function of the form $y=f(x)g(x)$ then:

$\dfrac{\mathrm d}{\mathrm dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$

The Quotient Rule: Given a function of the form $y=\dfrac{f(x)}{g(x)}$ then:

$\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{f(x)}{g(x)}\right]=\dfrac{g(x)f'(x)−f(x)g'(x)}{[g(x)]^2}$ 

It is useful to look at a few examples where these rules are applied.

Example 1: $y=3x^2\sin x$. Determine $\dfrac{\mathrm dy}{\mathrm dx}$

Explanation: It is important to understand that this function is the product of two functions with the first function $f(x)=3x^2$ and the second function $g(x)=\sin x$.

Therefore, since $\dfrac{\mathrm d}{\mathrm dx}(3x^2)=6x$ and $\dfrac{\mathrm d}{\mathrm dx}(\sin x)=\cos x$, then:

$\dfrac{\mathrm dy}{\mathrm dx}=6 x\sin x + 3x^2\cos x$

Note: You may see this in factored form as $\dfrac{\mathrm dy}{\mathrm dx}=3x(2\sin x+ x \cos x)$.

Example 2: 𝑦=𝑥3𝑥2+1 . Determine 𝑑𝑦𝑑𝑥 

Explanation: It is important to understand that this function is the quotient of two functions with the first function $f(x)=x^3$ and the second function $g(x)=x^2+1$.

Therefore, since $\dfrac{\mathrm d}{\mathrm dx}(x^3)=3x^2$ and $\dfrac{\mathrm d}{\mathrm dx}(x^2+1)=2x$, then:

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{(x^2+1)3x^2−x^3(2x)}{(x^2+1)^2}$ $= \dfrac{3x^4+3x^2−2x^4}{(x^2+1)^2}$ $= \dfrac{x^4+3x^2}{(x^2+1)^2}$