If y= (2x + 2)^3 find dy/dx using chains rule
Just use the power rule and the chain rule.
y = u^n
dy/dx = n u^(n-1) du/dx
Now just plug in your u=2x+2 and crank it out.
I might even go ahead and simplify it to
y = 2^3 (x+1)^3 = 8(x+1)^3
Then du/dx = 1
I would just do it in one step
dy/dx = 2(2x+3)^2 (2)
= 4(2x+3)^2
using the steps asked for:
let u = 2x+2
du/dx = 2
then y = u^3
dy/du = 3u^2
dy/dx = (dy/du)(du/dx) = (3u^2)(2)
= 4u^2
= 4(2x+2)^2
Once you see the pattern of these type of derivatives, you should be able to differentiate using the simple approach I used at the top.
To find the derivative of y = (2x + 2)^3 using the chain rule, we need to break down the function into its individual components and apply the chain rule step by step.
Step 1: Identify the outer function and the inner function.
In this case, the outer function is the cubing function f(u) = u^3, and the inner function is g(x) = 2x + 2.
Step 2: Find the derivatives of the inner and outer functions separately.
The derivative of the outer function f(u) = u^3 with respect to u is df/du = 3u^2.
The derivative of the inner function g(x) = 2x + 2 with respect to x is dg/dx = 2.
Step 3: Apply the chain rule formula.
The chain rule states that the derivative of the composite function y = f(g(x)) is given by dy/dx = df/du * dg/dx.
Step 4: Substitute the derivatives and the inner function into the chain rule formula.
dy/dx = df/du * dg/dx = (3u^2) * (2) = 6u^2.
Step 5: Substitute the inner function back into the chain rule formula.
Since the inner function is g(x) = 2x + 2, we substitute u = 2x + 2 into the chain rule formula.
dy/dx = 6(2x + 2)^2.
Therefore, the derivative of y = (2x + 2)^3 using the chain rule is dy/dx = 6(2x + 2)^2.