How do you find the derivative of u=(6+2x^2)^3u=(6+2x2)3?

2 Answers
Dec 30, 2017

(du)/(dx)=12x(6+2x^2)^2dudx=12x(6+2x2)2

Explanation:

we will need the chain rule

(du)/(dx)=color(red)((du)/(dt))color(blue)((dt)/(dx))dudx=dudtdtdx

u=(6+2x^2)^3u=(6+2x2)3

t=6+2x^2=>color(blue)((dt)/(dx)=4x)t=6+2x2dtdx=4x

u=t^3=>color(red)((du)/(dt)=3t^2)u=t3dudt=3t2

(du)/(dx)=color(red)(3t^2)xxcolor(blue)(4x)dudx=3t2×4x

substitute back and tidy up

(du)/(dx)=12x(6+2x^2)^2dudx=12x(6+2x2)2

Dec 30, 2017

(du)/dx=12x(6+2x^2)^2dudx=12x(6+2x2)2

Explanation:

"differentiate using the "color(blue)"chain rule"differentiate using the chain rule

"given "u=f(g(x)" then"given u=f(g(x) then

(du)/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"

u=(6+2x^2)^3

rArr(du)/dx=3(6+2x^2)^2xxd/dx(6+2x^2)

color(white)(rArr(du)/dx)=12x(6+2x^2)^2