How do find the derivative of #y = cos^2 (x^2 - 3x)#?

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Answer 1 · 2015-08-22T22:22:42

Use the chain rule twice.

The chain rule is useful whenever you have nested functions, meaning something that looks like this;

#y = f( g(x) )#

The chain rule tells us;

#d/(dx) f(g(x)) = f'(g(x))g'(x)#

In this case, we have three nested functions;

#y = f(g(h(x)))#

Where;

#f = g^2#
#g = cos(h)#
#h = x^2 - 3x#

Plug each function into the one above it to see that you get the original function. To solve, use the chain rule first on #f#.

#f'(g) = 2g*g'#

Now we use the chain rule again to solve for #g'#.

#g'(h) = -sin(h)*h'#

And when we solve for #h'# via power rule;

#h'(x) = 2x - 3#

Now just plug in #f#, #f'#, #g#, #g'#, #h#, and #h'# and you get;

#f'(x) = 2cos(x^2-3x)(-sin(x^2-3x)(2x-3))#

#=(6-4x)sin(x^2-3)cos(x^2-3)#