How do you differentiate #y=(6x^2 + 2x)^3#?

Question

Evelyn Agard · Accepted Answer

#dy/dx= 3(12x + 2)(6x^2 + 2x)^2#

Don't bother expanding, just use the chain rule. Let #u = 6x^2+2x#. Then #du = 12x + 2#. This also means that #y = u^3#., or #y = 3u^2#.

#dy/dx = (12x +2)3(6x^2 + 2x)^2#

#dy/dx= 3(12x + 2)(6x^2 + 2x)^2#

Hopefully this helps!

Emily Ammerman · Answer

#3(6x^2+2x)^2(12x+2)# Applying the chain rule we arrive at the consensus that #d/dx f(g(x)) = f'(g(x))*g'(x)# we identify the variables within our problem and define that #f(x) = x^3# and #f'(x)=3(x)^2# #g(x) = 6x^2+2x# combining the functions together with recieve the original function #g'(x)=12x+2# using the power rule #Px^(P-1)# from this point we plug out g(x) and f(x) functions into our rules of derivatives.

Christopher Allers · Answer

undefined To differentiate the function y=(6x^2 + 2x)^3, you can use the chain rule. The chain rule states that if you have a composite function, f(g(x)), then the derivative is f'(g(x)) times g'(x). Applying this to the given function:

1. Find the derivative of the outer function: 
   Let u = 6x^2 + 2x
   Then y = u^3
   dy/du = 3u^2

2. Find the derivative of the inner function:
   du/dx = d(6x^2 + 2x)/dx = 12x + 2

3. Multiply the derivatives found in steps 1 and 2:
   dy/dx = (dy/du) * (du/dx) = 3(6x^2 + 2x)^2 * (12x + 2)

So, the derivative of y=(6x^2 + 2x)^3 is dy/dx = 3(6x^2 + 2x)^2 * (12x + 2).