Find the derivative...?

Question

y= ∫ cos√(t) dt; from x^4 to 0

 

 

find dy/dx

Answer 1

Derivative of an Integral (Fundamental Theorem of Calculus):

Herer f(t) = cos√t so, f(x) = sin√x = y

y = sin√x

Apply derivatve each side.

Derivative of Trigonometric Functions:

                             [ inner derivation of √x is 1/(2√x) ]

Comments

The derivative of y= ∫ cos√(t) dt; from x^4 to 0 is

Answer 2

THE SECOND FUNDAMENTAL THEOREM OF CALCULUS : If f is continuous on an open interval I containing a, then, for every x in the interval, 

The integration function is .

Apply derivative each side with respect to x.

Let , and apply the Second Fundamental Theorem of Calculus with the Chain Rule as shown.

Apply Chain Rule :

Substitute .

Apply the Second Fundamental Theorem of Calculus :