What is the derivative of $({cos}^{2} (x) {sin}^{2} (x))$ $(cos^2(x)sin^2(x))$ ?

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What is the derivative of $({cos}^{2} (x) {sin}^{2} (x))$ $(cos^2(x)sin^2(x))$ ?

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Answer 1 · 2016-11-29T15:54:16

We could calculate the derivative using the product rule

$\frac{d (f \cdot g)}{d x} = \frac{d f}{d x} \cdot g + f \cdot \frac{d g}{d x}$ $(d(f*g))/(dx) = (df)/(dx)* g+f*(dg)/(dx)$

However we can also note that:

$sin (2 x) = 2 sin x cos x$ $sin(2x)=2sinxcosx$

so:

$f (x) = {cos}^{2} (x) {sin}^{2} (x) = {(\frac{sin 2 x}{2})}^{2} = \frac{1}{4} {sin}^{2} (2 x)$ $f(x) = cos^2(x)sin^2(x) = ((sin2x)/2)^2=1/4sin^2(2x)$

Using the chain rule:

$\frac{d (\frac{1}{4} {sin}^{2} (2 x))}{d x} = \frac{1}{4} \cdot 2 sin (2 x) cos (x) \cdot 2 = 2 sin x \cdot cos x \cdot cos x = 2 sin x {cos}^{2} x$ $(d(1/4sin^2(2x)))/dx= 1/4*2sin(2x)cos(x)* 2 = 2sinx * cos x * cos x= 2sinxcos^2x$

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What is the derivative of $({cos}^{2} (x) {sin}^{2} (x))$ $(cos^2(x)sin^2(x))$ ?

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What is the derivative of $({cos}^{2} (x) {sin}^{2} (x))$ $(cos^2(x)sin^2(x))$ ?