How do you differentiate #y = (x + 7)^10 (x^2 + 2)^7#?

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Answer 1 · 2015-07-06T11:32:03

#y'=(10(x^2+2)+14x(x+7))(x+7)^9(x^2+2)^6#

#= (24x^2+98x +20)(x+7)^9(x^2+2)^6#

#y=(x+7)^10(x^2+2)^7# is of the form:

#y=U(x)V(x)#

An equation of this form is differentaited like this:

#y'=U'(x)V(x)+U(x)V'(x)#

#U(x)# and #V(x)# are both of the form:

#U(x)=g(f(x))#

An equation of this form is differentiated like this:

#U'(x)=f'(x)g'(f(x))#

#rarr U'(x)=(d(x+7))/(dx)(d((x+7)^10))/(d(x+7))=1*10(x+7)^9#
#=10(x+7)^9#

#rarr V'(x)=(d(x^2+2))/(dx)(d((x^2+2)^7))/(d(x^2+2))=2x*7(x^2+2)^6#
#=14x(x^2+2)^6#

Therefore:

#y'=10(x+7)^9(x^2+2)^7+14x(x+7)^10(x^2+2)^6#

#=(10(x^2+2)+14x(x+7))(x+7)^9(x^2+2)^6#
#= (24x^2+98x +20)(x+7)^9(x^2+2)^6#