Q28P - Mathematical Methods in Physical Sciences

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.

Step-by-Step Solution

Answer

It is proved that the derivative is,

Step 1: 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, .

Step 2: 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, 

Step 3: 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.