How do you differentiate #f(x)=(5-x^2)(x^3-3x+3) # using the product rule?

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Answer 1 · 2015-12-27T00:18:40

#f'(x) = -5x^4 +24x^2 -6x-15#

Derivative of product rule

Given #" " " h= f*g#

#h' = fg' +f'g#

The original problem

#f(x) = (5-x^2)(x^3-3x+3)#

#f'(x) = (5-x^2) d/dx(x^3-3x+3) + d/dx(5-x^2)(x^3-3x+3)#

#=> (5-x^2)(3x^2-3) + (-2x)(x^3-3x+3)#

Now we can multiply and combine like terms

#=> (15x^2 -15 -3x^4 +3x^2) +( -2x^4+6x^2 -6x)#
#=> -5x^4 +24x^2 -6x-15#