Further chain rule examples

You have to know when to apply the chain rule. The chain rule can be applied to the following expressions.

an algebraic function of a trig function, e.g. $(1 + \sin x)^3$
${d \over {dx}}(1 + \sin x)^3 = 3(1 + \sin x)^2 (\cos x)$
(Differentiate the bracket cubed with respect to the bracket, then the bracket with respect to x)
a trigonometric function of an algebraic function, e.g. $\sin (3x^2 + 4)$
${d \over {dx}}(\sin (3x^2 + 4)) = \cos (3x^2 + 4)\;\times 6x$
(Differentiate the sine of the bracket with respect to the bracket, then the bracket with respect to x.)
a function of a function of a function of x, e.g. $\sin ^2 2x$
${d \over {dx}}(\sin ^2 2x) = {d \over {dx}}(\sin 2x)^2 = 2(\sin 2x)(\cos 2x)(2) = 2\sin 4x$
(Differentiate the bracket squared with respect to the bracket, then the bracket (sin2x) with respect to x.)
Note the use of sin2A = 2sinAcosA

It is not a good idea, especially if you intend to proceed with more mathematics, to learn the rule that ${d \over {dx}}(\sin ax) = a\cos ax$ , even though it appears in the formula list on the exam paper.

It's far better to think of the process as ${d \over {dx}}\;\;(\sin (ax)) = \cos (ax)\;\times a$ , in other words, differentiate the sine of (ax) with respect to (ax), then (ax) with respect to x.