Question

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

28.. (See hint below.)

Problem 28 is the chin rule for the derivative of a function of a function.

Short answer

It is proved that the derivative is,

Step-by-step solution

To write the definition (2.1) of derivative of function

The definition of the derivative of the functionis as follows:

The derivative of the function is defined as,

Where, .

To write the composition of two functions

Let and be two differentiable functions.

Let be the function.

By using the definition of differentiation,

Thus,

Now,

Next,

To find the derivative and prove the result

Since both the functions are differentiable, writing under the same limit, the above equation becomes,

.

Hence, it is proved.