One application of the chain rule for differentiation is differentiating inverse functions. This will help you find the derivative of functions you don't know how to differentiate directly, but that you do know how to differentiate the function's inverse.
Inverse function reminder
Let f(x) be a function. Then its inverse is denoted by f−1(x). Examples of functions and their inverse includes ex and ln(x), x2 and x, and sin(x) and sin−1(x).
Example 1
Find the inverse function of g(x)=4x−3.
To find the inverse function, rearrange to make x the subject:
x=4g(x)+3
Next, replace the x on the left with g−1(x) and replace g(x) with x:
g−1(x)=4x+3
Using the chain rule
The chain rule essentially gives you a method to manipulate derivatives. For example, it says that
dxdy=dudy×dxdu
You can approach statements like this as you would to a fraction. Notice that it is as if you are cancelling the du.
Note: It is not actually as straightforward as simply cancelling, but this is a helpful way to look at it.
For y=f(x), you have that the derivative of f is dxdy.
The inverse is given by x=f−1(y) and thus the derivative of the inverse function is dydx.
Now consider that by the chain rule, the following is true:
1=dxdy×dydx
Thus you have the following result:
dydx=dxdy1
This is the derivative of the inverse function. Equivalently:
dxdy=dydx1
Example 2
Using the derivative of the inverse function rule, show that dxd(ln(x))=x1.
Let y=ln(x). You seek dxdy. The function can be re-expressed as ey=x. This is the inverse function and you know how to differentiate it:
dydx=ey
Thus:
dxdy=dydx1=ey1=x1
Example 3
Use the derivative of the inverse function rule to differentiate sin−1(x) with respect to x.
Let y=sin−1(x). You seek dxdy. The function can be rearranged to give x=sin(y). This is the inverse function. Thus:
dydx=cos(y)
Hence
dxdy=cos(y)1
Now you need a way to express cos(y) in terms of x. To do this, use that x=sin(y) and the identity 1=sin2(y)+cos2(y):
