Differentitate y= x^2(sqroot4x-3). Do not simplify. Reads: x squared times the square root of (4x minus 3)

To differentiate the function y = x^2(sqrt(4x - 3)), we can use the product rule. The product rule states that if we have a function that is the product of two other functions, we can differentiate it by keeping one function as it is and differentiating the other, then adding these two differentiations together.

Let's break down the given function:
f(x) = x^2 * sqrt(4x - 3)

In this case, we have two functions being multiplied: f(x) = u(x) * v(x), where u(x) = x^2 and v(x) = sqrt(4x - 3).

Now, let's differentiate each function separately:
u'(x) = d/dx(x^2) = 2x

v'(x) requires the chain rule because we have a composition of functions. The chain rule states that if we have a function within another function, we should first differentiate the outer function, and then multiply it by the derivative of the inner function.
Let's define g(x) = 4x - 3, so v(x) = sqrt(g(x)). Applying the chain rule:
v'(x) = d/dx(sqrt(g(x))) * g'(x)

To differentiate sqrt(g(x)), we need to differentiate the square root function, which is 1/2 multiplied by g'(x) (the derivative of the inner function):
v'(x) = (1/2) * (4) * g'(x) = 2g'(x)

Since g(x) = 4x - 3, its derivative g'(x) is 4.

Finally, we can put everything together to get the derivative of f(x):
f'(x) = u'(x) * v(x) + u(x) * v'(x)
= (2x) * sqrt(4x - 3) + (x^2) * 2(4)
= 2x sqrt(4x - 3) + 8x^2

Therefore, the derivative of y = x^2(sqrt(4x - 3)) is f'(x) = 2x sqrt(4x - 3) + 8x^2.