Use 2nd Fundamental Theorem of Calculus to find derivative of f(x) = integral of 2x^2 (at the top) to x-5 (at the bottom) of square root of Sin(x)dx

Hmmmm. You want the derivative of the integral.

well the integral is -cos(2x^2)+cos(x-5)
and the derivative of that is

sin(2x^2)*4x -cos(x-5)

To find the derivative of the function f(x) = integral from x-5 to x of square root of Sin(x) dx, we can utilize the 2nd Fundamental Theorem of Calculus.

The 2nd Fundamental Theorem of Calculus states that if a function f(x) is continuous on the closed interval [a, b] and F(x) is an antiderivative of f(x) on that interval, then the integral of f(x) from a to b equals F(b) - F(a). This theorem allows us to evaluate the derivative of a function that involves an integral.

Let's proceed with finding the derivative of f(x):

Step 1: Apply the 1st Fundamental Theorem of Calculus to find an antiderivative of the integrand.
The antiderivative of the square root of Sin(x) with respect to x is denoted as F(x), so F(x) = ∫√(Sin(x)) dx.

Step 2: Apply the 2nd Fundamental Theorem of Calculus.
The derivative of f(x) is given by d/dx (F(x) - F(x-5)).

Step 3: Apply the chain rule and evaluate the derivative.
Using the chain rule, we find that d/dx (F(x) - F(x-5)) = F'(x) - F'(x-5).
Thus, the derivative of f(x) is given by F'(x) - F'(x-5).

Step 4: Compute the derivatives.
To find F'(x), we differentiate F(x) with respect to x.
Similarly, to find F'(x-5), we differentiate F(x) with respect to (x-5).

Step 5: Substitute the antiderivative, and its derivative, into the expression.
Replace F'(x) and F'(x-5) with their respective derivatives to obtain the final expression for the derivative of f(x).

Note: The process of finding the antiderivative and its derivative may involve various calculus techniques such as substitution, integration by parts, or trigonometric identities. These steps would vary depending on the specific integrand.

It is important to follow these steps carefully to ensure accurate calculation.