3. Derivatives of composite functions

Derivatives of composite functions

Theorem:  (Chain Rule) Let f be a real valued function which is a composite of two

functions u and v; i.e., f = v o u. Suppose t = u(x) and if both  

we have  

Example: Find the derivative of the function given by f (x) = sin (x²).

Ans:

Let

Alternatively, We can also directly proceed as follows:

Example :

Find the derivative of sin x – cos x.

Solution:

Given function is: sin x – cos x

Let f(x) = sin x and g(x) = cos x

Using the difference rule of differentiation,

d/dx [f(x) – g(x)] = d/dx f(x) – d/dx g(x)

d/dx (sin x – cos x) = d/dx (sin x) – d/dx (cos x)

= cos x – (-sin x)

= cos x + sin x

Problem:

Differentiate the functions with respect to x.

 2cot⁡(x2)

Ans;