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

Question

Scarlett Allison · Accepted Answer

#f'(x)=-2sinx(sinx+2xcosx)#

Product rule: for a function

#f(x)=g(x)h(x)#,

#f'(x)=g'(x)h(x)+h'(x)g(x)#

So,

#g(x)=-2x# #h(x)=sin^2x#

Find the derivative of each.

#g'(x)=-2#

Use the chain rule to find #h'(x)#, recalling that #d/dx u^2=2u*(du)/dx#.

Thus,

#h'(x)=2sinxd/dxsinx=2sinxcosx#

Put this all together:

#f'(x)=-2sin^2x+(-2x)2sinxcosx#

Simplify.

#f'(x)=-2sinx(sinx+2xcosx)#

Charles Anctil · Answer

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 #sin^2(x)#, so let's check that out first:

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

If we let #u(x) = sinx#, then we should get #2u * (du)/(dx)# like so:

#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)#

Scarlett Allison · Answer

undefined To differentiate $ f(x) = -2x\sin^2(x) $ using the product rule:

1. Identify the two functions being multiplied together: $ u(x) = -2x $ and $ v(x) = \sin^2(x) $.
2. Apply the product rule formula: $ (uv)' = u'v + uv' $.
3. Find the derivatives of $ u(x) $ and $ v(x) $.
   - $ u'(x) = -2 $
   - $ v'(x) = 2\sin(x)\cos(x) $ (using the chain rule for differentiation)
4. Substitute the derivatives into the product rule formula.
   - $ (uv)' = (-2)(\sin^2(x)) + (-2x)(2\sin(x)\cos(x)) $
5. Simplify the expression.
   - $ f'(x) = -2\sin^2(x) - 4x\sin(x)\cos(x) $