How do you use the product Rule to find the derivative of #y (x) = (x^2+1)(x^3+1)#?

2 Answers
Gió
Jul 27, 2015

I found: #y'(x)=5x^4+3x^2+2x#

Explanation:

The product rule allows you to derive a function formed by the product of two functions as #f(x)=g(x)*h(x)# to get:
#f'(x)=g'(x)h(x)+g(x)h'(x)#
In your case you have (in red are the derivatives):
#y'(x)=color(red)(2x)(x^3+1)+(x^2+1)color(red)(3x^2)=#
#=2x^4+2x+3x^4+3x^2=5x^4+3x^2+2x#

Truong-Son N.
Jul 27, 2015

You can always multiply it out if you don't want to use the product rule.

Product Rule:
#d/(dx)[f(x)g(x)] = f(x)(dg(x))/(dx) + g(x)(df(x))/(dx)#

#= (x^2 + 1)(3x^2) + (x^3+1)(2x)#

#= (3x^4 + 3x^2) + (2x^4+2x)#

#= color(blue)(5x^4 + 3x^2 + 2x)#

You can also do this with the Power Rule:

#h(x) = (x^2+1)(x^3+1)#

#h(x) = x^5 + x^2 + x^3 + 1#

#d/(dx)[h(x)]#

#=> 5x^4 + 2x + 3x^2#

#= color(blue)(5x^4 + 3x^2 + 2x)#

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