How do you use the chain rule to differentiate #y=4(x^3+1)^(1/2)#?

Question

Charles Anctil · Accepted Answer

#(6x^2)/(x^3+1)^(1/2)#

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

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|))).... (A)#

let #u=x^3+1rArr(du)/(dx)=3x^2#

and #y=4u^(1/2)rArr(dy)/(du)=2u^(-1/2)#

substitute these values into (A) and change u into terms of x.

#dy/dx=2u^(-1/2)xx3x^2=(6x^2)/(x^3+1)^(1/2)#

Charles Anctil · Answer

undefined To differentiate $ y = 4(x^3 + 1)^{1/2} $ using the chain rule, follow these steps:

1. Identify the outer function $ u $ and the inner function $ v $.
   $ u = 4v^{1/2} $
   $ v = x^3 + 1 $

2. Find the derivatives of $ u $ and $ v $ with respect to $ x $.
   $ \frac{du}{dv} = 2v^{-1/2} $
   $ \frac{dv}{dx} = 3x^2 $

3. Apply the chain rule:
   $ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} $

4. Substitute the derivatives into the chain rule equation:
   $ \frac{dy}{dx} = 2(x^3 + 1)^{-1/2} \cdot 3x^2 $

5. Simplify the expression if necessary:
   $ \frac{dy}{dx} = 6x^2(x^3 + 1)^{-1/2} $

That's the derivative of $ y = 4(x^3 + 1)^{1/2} $ using the chain rule.