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Calculus and differentiating

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2 Answers

You substitute g(x) into the equation for f(x) in place of x
for f(x)= 8x3 + 5x2 - 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 ) = du2/du(du/dx) , u = 2x+7 , d/du(u2) = 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 ) = du3/du(du/dx) , u = 2x+7 , d/du(u3) = 3u2
 
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

Comments

Thank you so much! I have been struggling with this subject for quite some time. This is a huge help!!
f(g(x)) = 8g(x)3 + 5g(x)2 - 7
 
Let u = g(x)
 
f(g(x)) = f(u) = 8u3 + 5u2 - 7
 
To find the derivative of any composite function, use the chain rule:
 
f ' = df(g(x))/dx = df(u)/du * du/dx
 
f ' = (24u2 + 10u)*(2)
 
f ' = 48(2x+7)2 + 20(2x+7)
 
Simplify to get the answer