Find the derivative of y = 4^2x.

To find the derivative of y = 4^2x, we can use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then the derivative of that function is given by f'(g(x)) * g'(x).

In this case, our function is f(g(x)) = 4^2x. We can think of g(x) as 2x, and f(u) as 4^u.

First, let's find the derivative of f(u), which is 4^u. To do this, we can take the natural logarithm (ln) of both sides of the equation, since the derivative of ln(u) with respect to u is 1/u.

ln(y) = ln(4^2x)
ln(y) = 2x * ln(4)

Next, we can differentiate both sides of the equation with respect to x. Remember that y itself is a function of x.

d/dx (ln(y)) = d/dx (2x * ln(4))

To differentiate the left side, we use the chain rule again:
1/y * dy/dx = 2 * ln(4)

Finally, we solve for dy/dx:
dy/dx = y * 2 * ln(4)
dy/dx = 4^2x * 2 * ln(4)

Therefore, the derivative of y = 4^2x is dy/dx = 4^2x * 2 * ln(4).