How do you differentiate $f (x) = {(- 7 x^{2} - 5)}^{8} {(2 x^{2} - 9)}^{9}$ $f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9$ ?

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Answer 1 · 2016-05-07T04:07:28

$color(red)(f' (x)=4x(7x^2+5)^7*(2x^2-9)^8(119x^2-207))$

For the given function $f(x)=(-7x^2-5)^8 (2x^2-9)^9$ ,
we are going to use the formula $d/dx(uv)=ud/dx(v)+v*d/dx(u)$

Let $u=(-7x^2-5)^8=(-1)^8(7x^2+5)^8=(7x^2+5)^8$ and $v=(2x^2-9)^9$

$f' (x)=(7x^2+5)^8 d/dx(2x^2-9)^9+(2x^2-9)^9*d/dx(7x^2+5)^8$

$f' (x)=(7x^2+5)^8*9(2x^2-9)^(9-1)d/dx(2x^2-9)+(2x^2-9)^9*8(7x^2+5)^(8-1)d/dx(7x^2+5)$

$f' (x)=(7x^2+5)^8*9(2x^2-9)^8*(4x-0)+(2x^2-9)^9*8(7x^2+5)^(7)(14x+0)$

$f' (x)=(7x^2+5)^8*9(2x^2-9)^8*(4x)+(2x^2-9)^9*8(7x^2+5)^(7)(14x)$

Factoring common factors

$f' (x)=$
$4x(7x^2+5)^7*(2x^2-9)^8[(7x^2+5)*9+(2x^2-9)*2(14)]$

$f' (x)=4x(7x^2+5)^7*(2x^2-9)^8[(63x^2+45)+(56x^2-252)]$

$color(red)(f' (x)=4x(7x^2+5)^7*(2x^2-9)^8(119x^2-207))$

God bless....I hope the explanation is useful.

How do you differentiate f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9f(x)=(−7x2−5)8(2x2−9)9f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9 ?