How do you use the chain rule in differentiation?

The chain rule is used in differentiation when you have a function inside another function, called a composite function. It helps you find the derivative of the whole expression.

Formula:
If ( y = f(g(x)) ), then the derivative is:
[ frac{dy}{dx} = f'(g(x)) cdot g'(x) ]

Example:
Suppose ( y = sin(3x) ).

• The outer function is ( f(u) = sin(u) ); the inner is ( g(x) = 3x ).
• The derivative of the outer function with respect to the inner is ( cos(3x) ).
• The derivative of the inner function is ( 3 ).

So,
[ frac{dy}{dx} = cos(3x) cdot 3 = 3cos(3x) ]

Takeaway:

• Use the chain rule when differentiating composite functions.
• Multiply the derivative of the outer function (with inner plugged in) by the derivative of the inner function.