The Chain Rule | Mathematico

Let

f (x) = \sin (x)

and

g (x) = x^{2}

Recall that we can compose these functions to make a new function

f (g (x)) = \sin (x^{2})

Here is the curve

y = f (g (x))

How should we differentiate this function? We need something called the chain rule:

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)

When you look at the chain rule, it isn't clear what you actually need to do. An alternative way to state the chain rule is

\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}

which, as we will see, makes it a little easier to use to begin with.

Differentiate the functions