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We can solve y=(2+4x^2)^3 in two ways. One is use the chain rule
y' = 3*8x*(2 + 4x^2)^2
=> y' = 24x*( 4 + 16x^4 + 16x^2)
=> y' = 96x + 384x^5 + 384x^3
Else expand (2+4x^2)^3
=> 8 + 64x^6 + 96x^4 + 48x^2
y' = 384x^5 + 384x^3 + 96x
The derivative of y=(2+4x^2)^3 is 384x^5 + 384x^3 + 96x
Since the given function is composed, we'll apply chain rule to differentiate it.
We'll differentiate with respect to x. First, we'll identify the composed functions, whose final result is y.
dy/dx = (dy/dt)*(dt/dx)
We'll put 2+4x^2 = t
y = t^3
We'll differentiate with respect to t:
dy/dt = d(t^3)/dt
dy/dt = 3t^2
We'll differentiate t with respect to x.
dt/dx = d(2+4x^2)/dx
dt/dx = 8x
dy/dx = 3t^2*8x = 24x*t^2
We'll substitute back t:
dy/dx = 24x(2 + 4x^2)^2
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