# How do you differentiate # f(x)=-2xsin^2(x) # using the product rule?

##### 2 Answers

#### Answer:

#### Explanation:

Product rule: for a function

So,

Find the derivative of each.

Use the chain rule to find

Thus,

Put this all together:

Simplify.

I remember the product rule by remembering the phrase:

"

first, d-second, plus second, d-first"

or:

#color(green)(d/(dx)[g(x)h(x)] = g(x)(dh(x))/(dx) + h(x)(dg(x))/(dx))#

The derivative here requires that you use the chain rule on

#color(green)(d/(dx)[f(u(x))] = (df(u(x)))/cancel(du) * cancel(du)/(dx) = (df(u(x)))/(dx))#

If we let

#d/(dx)[(sinx)^2] = 2sinx * d/(dx)[sinx] = 2sinxcosx#

So, the final result is:

#d/(dx)[-2xsin^2x] = (-2x)(2sinxcosx) + (sin^2x)(-2)#

#= color(blue)(-4xsinxcosx - 2sin^2x)#