You substitute g(x) into the equation for f(x) in place of x

for f(x)= 8x^{3} + 5x^{2} - 7 and g(x)=
2x+7

F(g(x)) = 8(2x+7)^{3} + 5(2x+7)^{2} - 7

d/dx 8(2x+7)^{3} + 5(2x+7)^{2} - 7

differentiate the sum term by term and factor out constants

d/dx(-7) + 5[ d/dx( (7+2x)^{2} )] + 8[ d/dx( (7+2x)^{3} )]

derivative of 7 is zero, simplify

5[ d/dx( (7+2x)^{2} )] + 8[ d/dx( (7+2x)^{3} )]

use the chain rule where

d/dx( (7+2x)^{2} ) = du^{2}/du(du/dx) , u = 2x+7 , d/du(u^{2}) = 2u

5[ 2(2x+7) d/dx(7+2x)] + 8[ d/dx( (7+2x)^{3} )]

Simplify

10(7+2x) [ d/dx (7+2x)] + 8[ d/dx( (7+2x)^{3} )]

differentiate the sum term by term and factor out constants

10(2x+7) [ d/dx(7) +2
d/dx(x) ] + 8[ d/dx( (7+2x)^{3} )]

derivative of 7 is zero , derivative of x is 1, simplify

20(2x+7) + 8[ d/dx( (7+2x)^{3} )]

use the chain rule where

d/dx( (2x+7)^{3} ) = du^{3}/du(du/dx) , u = 2x+7 , d/du(u^{3}) = 3u^{2}

20(2x+7) + 8( 3(2x+7)^{2} [
d/dx( (7+2x)]

simplify, and differentiate the sum term by term and factor out constants

20(2x+7) + 24(2x+7)^{2} [ d/dx(7) +
2d/dx(x) ]

derivative of 7 is 0, derivative of x is 1, simplify x2

20(2x+7) + 48(2x+7)^{2}

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