Derivative of a Function of a Function

Ask a Question, Get an Answer!

Type your question here

Hundreds of tutors are online and ready to help you right now!

So far, we know how to differentiate functions like sin x and x³- 5. But how do we differentiate a function of a function? That is how can we differentiate sin (x³- 5)?So far, we know how to differentiate functions like sin x and x³- 5. But how do we differentiate a function of a function? That is how can we differentiate sin (x³- 5)?

This differentiation of function of a function is known as 'chain rule'. The chain rule is probably the most widely used differentiation rule in mathematics.

We state the chain rule as follows:

If y=f(u) and u=g(x) are differentiable functions of u and x respectively, then

In other words,

if y is a differentiable function of u and u is a differentiable function of x, then the derivative of y with respect to x is equal to the derivative of y with respect to u times the derivative of u with respect to x.

Example 1:

Suggested answer:

Let y = log u, u = tan t, t = x/2

From the above example, it is clear that chain rule can be extended.

Let v = f(u), u = g(x), x = h(t)

Then, we have

Example 2:

Suggested answer:

Let Dx be the increment of x and Du be the corresponding increment in u.

or

or

Consider the function y = e^u.

Let Dy be the corresponding increment in y for the increment Du in u.

= e^u(1) = e^u ……..(2)

Therefore, we get