Here is a quick lesson covering the Product Rule, and the Quotient Rule in Calculus.
Prerequisite
This lesson assumes that the student knows what a calculus derivative is and is familiar with derivatives of functions such as \(x^2\), \(\sin(x)\) and so on.
The product rule is used when you want to take the derivative of one function \(f(x)\) multiplied by another (different) \(g(x)\) function. We have:
\[h(x) = f(x) g(x)\]
The derivative of \(h(x)\) is \(h'(x)\) and the form is as follows:
\[h'(x) = f'(x) g(x) + f(x) g'(x)\]
Think of the derivative \(h'(x)\) as shown above as:
\[ \text{(derivative of the first times the second function)} + \text{(first function times the derivative of the second function)}\]
Example of Product Rule
Suppose that \(h(x) = x \cos(x)\) and \(h(x)\) is in the form of \(h(x) = f(x) g(x)\).
Here we have \(f(x) = x\) with \(f'(x) = 1\) and \(g(x) = \cos(x)\) with \(g'(x) = - \sin(x)\).
Substituting the components into the product rule formula in (1.2) would give \(h'(x)\) as follows:
\[h'(x) = \cos(x) - x \sin(x)\]
The quotient rule is similar to the product rule but it has more steps. It is used when you want to take the derivative of one function \(f(x)\) divided by another (different) \(g(x)\) function. We have:
\[h(x) = \dfrac{f(x)}{g(x)}\]
The derivative of h(x) is h’(x) and the form is as follows:
\[h'(x) = \dfrac{f'(x) g(x) - f(x) g'(x)}{(g'(x))^2}\]
This time, think of the derivative \(h'(x)\) in (1.2) as:
\[\dfrac{\text{(derivative of the first times the second function)} - \text{(first function times the derivative of the second function)}}{\text{second function squared}}\]
Example of Quotient Rule
Suppose that \(h(x) = \dfrac{\sin(x)}{x^{2}}\) and \(h(x)\) is in the form of \(h(x) = \dfrac{f(x)}{g(x)}\).
Here we have \(f(x) = \sin(x)\) with \(f'(x) = \cos(x)\) and \(g(x) = x^{2}\) with \(g'(x) = 2x\) and \((g(x))^{2} = (x^{2})^{2}= x^{4}\).
Substituting the components into the quotient rule formula in with some factoring and simplifying would yield:
\[h'(x) = \dfrac{x \cos(x) - 2\sin(x)}{x^{3}} \text{ or } h'(x) = \dfrac{\cos(x)}{x^{2}} - \dfrac {2\sin(x)}{x^{3}}\].
I personally use this memory trick often for both product rule and quotient rule. Instead of using the argument \(x\) in \(f(x)\), I use just \(f\).
Instead of \(h'(x) = f'(x) g(x) + f(x) g'(x)\), an easier way is to use \(h' = f' g + fg'\).
For the quotient rule it would be: \(h' = \dfrac{f'g - f g'}{(g')^2}\).
Note that when using the shorthand notation, one should know what the independent variable is. In this case, it would be \(x\).