## the chain rule

first consider a problem:

sin'(2x)=?

if you are familiar with trigonometry, sin(2x) shall be immediately converted to 2sin(x)cos(x). then we can easily crack up the problem using the product rule. here, we present a much faster way:

the derivative of f(g(x)) is equal to f'(g(x))*g'(x).

so, for sin(2x). what is its derivative? cos(2x)*(2x)'. (sometimes, i get lazy, and represent the derivative of g(x)=2x by (2x)') This makes it 2cos(2x). as simple as that. you can apply the product rule as mentioned above and confirm this result. if you got it wrong, reconfirm.

sin'(2x)=?

if you are familiar with trigonometry, sin(2x) shall be immediately converted to 2sin(x)cos(x). then we can easily crack up the problem using the product rule. here, we present a much faster way:

the derivative of f(g(x)) is equal to f'(g(x))*g'(x).

*or, it is equal to the derivative of the outer function evaluated at the inner functions times the derivative of the inner function.*__MEMORIZE!__so, for sin(2x). what is its derivative? cos(2x)*(2x)'. (sometimes, i get lazy, and represent the derivative of g(x)=2x by (2x)') This makes it 2cos(2x). as simple as that. you can apply the product rule as mentioned above and confirm this result. if you got it wrong, reconfirm.