How do you find the derivative of the trig function #(cos^2(x)sin^2(x))#?

Question

Aurora Alvarado · Accepted Answer

Firstly find the derivates of the functions #ln(cosx)# and #ln(sinx)# as we'll be using them to find the derivative of the trig function: #cos^2xsin^2x# using logarithmic implicit differentiation .

As you know the derivatives of these two isolated functions you can now work towards finding the derivative of #cos^2xsin^2x#...

Evelyn Agard · Answer

undefined To find the derivative of the trigonometric function $ \cos^2(x) \sin^2(x) $, you can use the product rule followed by chain rule:

Let $ u = \cos^2(x) $ and $ v = \sin^2(x) $.

Then, $ \frac{du}{dx} = 2\cos(x)(-\sin(x)) $ and $ \frac{dv}{dx} = 2\sin(x)\cos(x) $.

Applying the product rule:

$ \frac{d}{dx}[\cos^2(x)\sin^2(x)] = \frac{du}{dx}v + u\frac{dv}{dx} $

Substitute the derivatives:

$ = (2\cos(x)(-\sin(x)))\sin^2(x) + \cos^2(x)(2\sin(x)\cos(x)) $

Simplify:

$ = -2\cos(x)\sin^3(x) + 2\sin(x)\cos^3(x) $

Thus, the derivative of $ \cos^2(x)\sin^2(x) $ is $ -2\cos(x)\sin^3(x) + 2\sin(x)\cos^3(x) $.