Question

What is the product, quotient, and the chain rule?
How are they suppose to help us calculate a derivative?

Answer 1

They're handy rules for finding the derivative of complicated functions.

The product rule is used for the product of functions, like \(f(x)g(x)h(x)\).

The quotient rule is used for rational functions, which look like \(\dfrac{f(x)}{g(x)}\).

The chain rule is used for compositions of functions, like \((f\circ g)(x)=f(g(x))\).

Using my general examples, the derivatives of each function is as follows:
\[\frac{d}{dx}[f(x)g(x)h(x)]=\\\left(\frac{d}{dx}[f(x)]\right)g(x)h(x)+f(x)\left(\frac{d}{dx}[g(x)]\right)h(x)+f(x)g(x)\left(\frac{d}{dx}[h(x)]\right)\]
(Note: the number of terms on the RHS is equal to the number of functions you're multiplying together, but not necessarily the number of terms in the actual derivative function.)

\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{\left(\left(\dfrac{d}{dx}[f(x)]\right)g(x)-f(x)\left(\dfrac{d}{dx}[g(x)]\right)\right)}{\left[g(x)\right]^2}\]

\[\frac{d}{dx}[f(g(x))]=\frac{d}{dx}[f(g(x))]\cdot\frac{d}{dx}[g(x)]\]
It may be clearer to see if I used Lagrange's notation (prime notation):
\[[f(g(x))]'=f'(g(x))\cdot g'(x)\]