Solve using chain rule

y=(3x^3+1)(-4x^2-3)^4

So far, I have:

y'=(3x^3+1)*4(-4x^2-3)^3*(-8x)+(-4x^2-3)^4*(9x^2)

fine so far

To solve this problem using the chain rule, we need to consider the fact that we have a composition of functions. The outer function is the power function, and the inner function is the product function.

Let's start by finding the derivative of the outer function, which is the power function. Since we are raising the expression (-4x^2 - 3) to the power of 4, its derivative will be a constant multiple of the expression (-4x^2 - 3) raised to the power of 3.

Next, we need to find the derivative of the inner function, which is the product of two functions: 3x^3 + 1 and (-4x^2 - 3). To do this, we use the product rule.

The product rule states that if we have two functions u(x) and v(x), the derivative of their product, u(x) * v(x), is given by:

(u * v)' = u' * v + u * v'

Let's find the derivatives of the functions u(x) = 3x^3 + 1 and v(x) = (-4x^2 - 3).

u'(x) = 9x^2
v'(x) = -8x

Now, we can use the product rule to find the derivative of the inner function:

(u * v)' = (3x^3 + 1) * (-8x) + (9x^2) * (-4x^2 - 3)

Expanding this expression, we get:

(u * v)' = -24x^4 - 36x^2 - 8x + 36x^4 + 12x^2

Combining like terms, we have:

(u * v)' = 12x^4 - 24x + 12x^2

Finally, we can combine the derivatives of the outer and inner functions to find the derivative of the original function:

y' = (3x^3 + 1) * 4(-4x^2 - 3)^3 * (-8x) + (-4x^2 - 3)^4 * (12x^4 - 24x + 12x^2)

And that's the derivative of the function y = (3x^3 + 1)(-4x^2 - 3)^4, using the chain rule and the product rule.