Compute the derivative of the given function below

Question

y= ⁰∫_√x sin(t²) dt

dy/dx=?

Answer 1

Let u = √x

dy/dx = (dy/du)(du/dx) (chain rule)

(Fundamental Theorem of Calculus)

plugging in u = √x we get

Answer 2

Use the fundamental theorem of calculus: Assuming the lower limit of integration is 0, and the upper limit of integration is sqrt(x):

dy/dx = sin( sqrt(x)^2 ) * 1/(2sqrt(x))