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;

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

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

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