Find the derivative of the function.

Question

(a) Find the derivative of the function g (x) = sin2 x + cos2 x in two ways.

(b) For f (x) = sec2 x and g (x) = tan2 x, show that f' (x) = g' (x).

Answer 1

Step 1:

(a)

The function is .

The derivative of the function can be found in two ways.

One is differentiating the function directly and then simplifying.

Other is by simplifying the function and differentiating the simplified function.

Apply derivative on each side.

Apply chain rule of derivative: .

.

Step 2:

The function is .

Trigonometric identities : .

Apply derivative on each side.

Derivative of a constant is always 0.

.

Solution :

(a) .

Answer 2

Step 1:

(b)

The functions are and .

Consider .

Apply derivative on each side.

Apply chain rule of derivative: .

Step 2:

Consider .

Apply chain rule of derivative: .

Substitute in the above expression.

.

Solution :

(b) .