(a) \ \

Step 1 : \ \

The fundamental theorem of calculus, part 1 :

If  is continuous on then the function  is defined by is continuous on and differentiable on , and .

Step 2 :

The equation is

Compare with

The function .

Thus, from the fundamental theorem of calculus, part 1

The derivative of the function is,

. \ \

The derivative of the function

Solution :

The derivative of the function .

1. (b)

Step 1 : \ \

The fundamental theorem of calculus, part 1 :

If  is continuous on then the function g is defined by is continuous on and differentiable on , and .

Step 2 :

The equation is .

Apply definition of special definite integrals: .

.

.

The derivative of the function .

Solution :

The derivative of the function .

1. (c)

Step 1 : \ \

The fundamental theorem of calculus, part 1 :

If  is continuous on then the function g is defined by is continuous on and differentiable on , and .

Step 2 :

The equation is .

Apply Additive interval property: .

Apply definition of special definite integrals: .

Apply derivative on each side.

The derivative of the function .

Solution :

The derivative of the function .