How do you differentiate #g(x) = (2sinx -e^x) ( cosx-x^2)# using the product rule?

Question

Luke Alvarez · Accepted Answer

#g'(x)=2cos^2(x)-e^xcos(x)-2x^2cos(x)+x^2e^x-2sin^2(x)+e^xsin(x)-4xsin(x)+2xe^x# using the product rule we get #g'(x)=(2cos(x)-e^x)(cos(x)-x^2)+(2sin(x)-e^x)(-sin(x)-2x)#

Evelyn Agard · Answer

undefined To differentiate g(x) = (2sinx - e^x) (cosx - x^2) using the product rule, follow these steps:

1. Identify the functions u(x) = 2sinx - e^x and v(x) = cosx - x^2.
2. Apply the product rule: g'(x) = u'(x)v(x) + u(x)v'(x).
3. Differentiate u(x) and v(x) separately.
   - u'(x) = (2cosx - e^x)
   - v'(x) = (-sinx - 2x)
4. Substitute the values of u'(x), v'(x), u(x), and v(x) into the product rule formula.
   - g'(x) = (2cosx - e^x)(cosx - x^2) + (2sinx - e^x)(-sinx - 2x)
5. Simplify the expression if necessary.

So, the derivative of g(x) with respect to x is:

g'(x) = (2cosx - e^x)(cosx - x^2) + (2sinx - e^x)(-sinx - 2x)