The function is and where f and g have all order of derivatives.

(a) .

Apply first derivative on each side with respect to .

Product rule of derivatives: .

.

Apply second derivative on each side with respect to .

Therefore, .

(b)

.

Apply third derivative on each side with respect to .

.

Apply fourth derivative on each side with respect to .

(c)

Rewrite the second, third and fourth derivatives as follows

.

.

.

Exponent in above expressions represent the order of the derivative.

Observe the above pattern and coefficients of the each expression.

We will come to know that each derivative expression is replica of binomial expansion .

Using the binomial expansion we can guess the derivative of the function.

.

Where is derivative of the function .

Where is derivative of the function .

(a) .

(b) and .

(c) .