Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
If f(x)= ∫(4,x^3)√(t^2+10) dt then f '(x).
To find the derivative of the function f(x), we can use the Part 1 of the Fundamental Theorem of Calculus, which states that if a function F(x) is defined as the integral of another function f(t) from a constant 'a' to 'x', then the derivative of F(x) with respect to x is equal to f(x).
In this case, we have the function f(x) defined as the integral from 4 to x^3 of the function √(t^2 + 10) dt.
To find f '(x), we need to find the derivative of f(x) with respect to x. Let's proceed step by step:
1. Write down the function f(x):
f(x) = ∫(4, x^3)√(t^2 + 10) dt
2. Apply the Fundamental Theorem of Calculus, Part 1:
f '(x) = (√(x^3^2 + 10))'
3. Differentiate the function inside the square root:
f '(x) = (1/2) * (x^3^2 + 10)^(-1/2) * (2x^3)
4. Simplify the expression:
f '(x) = x^3 / √(x^3^2 + 10)
Therefore, the derivative of the function f(x) = ∫(4,x^3)√(t^2 + 10) dt is f '(x) = x^3 / √(x^3^2 + 10).