Home > Calculus > Derivative Rules for y=cos(x) and y=tan(x)

What is the derivative of #sin^2(x)cos^2(x)#?

Answer 1

Christopher Allers

#(sin 4x)/2#

#sin(x)cos(x) = (sin 2x)/2# #\implies sin^2(x)cos^2(x) = (sin^2 2x)/4#

#((sin^2 2x)/4)' = 1/4 (2 sin 2x cos 2x (2) ) = sin 2x cos 2x = (sin 4x)/2#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Paisley Johnson

To find the derivative of ( \sin^2(x) \cos^2(x) ), you can use the product rule. The product rule states that if ( f(x) ) and ( g(x) ) are differentiable functions, then the derivative of their product is given by ( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) ). Applying this rule to ( \sin^2(x) \cos^2(x) ), where ( f(x) = \sin^2(x) ) and ( g(x) = \cos^2(x) ), we get:

[ \frac{d}{dx}(\sin^2(x) \cos^2(x)) = (2\sin(x)\cos(x) \cdot \cos^2(x)) + (\sin^2(x) \cdot -2\sin(x)\cos(x)) ]

[ = 2\sin(x)\cos(x)\cos^2(x) - 2\sin^2(x)\cos(x) ]

[ = 2\sin(x)\cos(x)[\cos^2(x) - \sin^2(x)] ]

[ = 2\sin(x)\cos(x)\cos(2x) ]

So, the derivative of ( \sin^2(x) \cos^2(x) ) with respect to ( x ) is ( 2\sin(x)\cos(x)\cos(2x) ).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.