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:

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

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

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