What is the derivative of ${cos}^{4} (x) - {sin}^{4} (x)$ $cos^4(x)-sin^4(x)$ ?

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Answer 1 · 2016-08-30T02:11:25

$f'(x) = -2sin2x$

$f(x) = cos^4x-sin^4x$

First let's do some simplification.

Notice: $f(x) = (cos^2x+sin^2x)(cos^2x-sin^2x)$

Since $(cos^2x+sin^2x) = 1 -> f(x) = (cos^2x-sin^2x)$

Applying the identity $cos(2x) = cos^2x-sin^2x$

$f(x) = cos(2x)$

$f'(x) = -2sin(2x)$ (Chain rule)

What is the derivative of cos^4(x)-sin^4(x)cos4(x)−sin4(x)cos^4(x)-sin^4(x)?