Differentiation by Substitution

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Differentiation of certain functions seem to be very difficult, but by suitably substituting the independent variable with some trigonometric function or other functions, they can be differentiated easily.Differentiation of certain functions seem to be very difficult, but by suitably substituting the independent variable with some trigonometric function or other functions, they can be differentiated easily.

If f(x) involves inverse trigonometric functions of algebraic functions, the following substitutions simplify the function f(x) to be differentiated.

Example 1:

Differentiate

Suggested answer:

Example 2:

Suggested answer:

Dividing the numerator and the denominator by cosq , we have

Differentiating with respect to x, we have