Calculus and differentiating

Question

calculate d/dx f(g(x)) is f(x)= 8x^3+5x^2-7 and g(x)= 2x+7

Answer 1

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

Answer 2

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