How do you differentiate #f(x)=(cosx+1)(-x^2-3e^x)# using the product rule?

Question

Ezra Adams · Accepted Answer

#(cosx + 1 )(-2x-3e^x) - sinx(x^2-3e^x)#

using the #color(blue)" Product rule "#

f'(x) = g(x).h'(x) + h(x).g'(x) is the result if f(x) = g(x).h(x)

#"----------------------------------------------------------------"#

here g(x) #= cosx + 1 rArr g'(x) = -sinx #

and #h(x) = (-x^2-3e^x) rArr h'(x) = -2x-3e^x # #"-----------------------------------------------------------------"#

now enter these outcomes in f'(x)

f'(x) #=(cosx+1)(-2x-3e^x)-sinx(x^2-3e^x)#

Evelyn Agard · Answer

undefined To differentiate $ f(x) = (cosx + 1)(-x^2 - 3e^x) $ using the product rule:

1. Identify the two functions being multiplied: $ u(x) = \cos(x) + 1 $ and $ v(x) = -x^2 - 3e^x $.
2. Apply the product rule: $ f'(x) = u'(x)v(x) + u(x)v'(x) $.
3. Differentiate $ u(x) $ and $ v(x) $ separately.
   - $ u'(x) = -\sin(x) $
   - $ v'(x) = -2x - 3e^x $
4. Substitute the derivatives and original functions into the product rule formula.
5. Compute $ f'(x) $.