Derivative of Inverse Function
Examples with detailed solutions on how to find the derivative (Differentiation) of an inverse function are presented. Also more exercises with answers are included.
Derivative of Inverse Function Formula (theorem)
Let f be a function and f−1 its inverse. One of the properties of the inverse function is that f(f−1(x))=x
Let y=f−1(x) so that.
f(y)=x
Differentiate both sides using chain rule on the left side.
dfdydydx=1
Solve for dydx
dydx=df−1dx=1dfdy
which may also be written as
df−1dx=1f′(f−1(x))
where f′ is the first derivative of f.
Example 1
Find the derivative of the inverse of function f given by f(x)=x2−1Solution to Example 1
We present two methods to answer the above question. In the first method we calculate the inverse function and then its derivative. In the second method, we use the formula developed above.Method 1
The first method consists in finding the inverse of function f and differentiate it. To find the inverse of f we first write it as an equation y=x2−1
Solve for x.
x=2y+2
Interchange x and y to obtain the inverse.
y=f−1(x)=2x+2
The above gives the inverse function of f whose derivative is given by
dydx=df−1dx=2
Method 2
The second method starts with one of the most important properties of inverse functions.Given f(x)=x2−1
hence
f′(x)=12
Substitute f′ by 12 in the formula df−1dx=1f′(f−1(x)) to obtain df−1dx=112=2
Note that The first method can be used only if we can find the inverse function explicitly.
Example 2
Find the derivative dydx where y=arcsinx.Solution to Example 2
arcsinx is the inverse function of sinx and hencesin(arcsin(x))=x(I)
Given y=arcsinx
Take the sine of both sides in the above
siny=sin(arcsinx)
Simplify using (I)
siny=x
Differentiate both sides of the above equation, with respect to x , using the chain rule on the left side.
dydxcosy=1
Solve for dydx
dydx=1cosy
Make the substitution y=arcsinx in the above
dydx=1cos(arcsinx)(II)
Simplify cosarcsinx as using the the trigonometric identity cosx=√1−sin2x by writing
cos(arcsinx)=√1−sin2(arcsinx)
Simplify using the property of inverse fiunctions: sin(arcsinx)=x
which gives
cos(arcsinx)=√1−x2
Substitute in (II) above to obtain the final answer
dydx=d(arcsin(x)dx=1√1−x2
Note that the above result could have been obtained using the formula (theorem) above but here we have shown how to find the derivative of the inverse without ( remembering ) the formula.
Exercises
Find the derivative of the inverse of each function given below.
- f(x)=3x−4
- g(x)=arccosx
- h(x)=arctanx
Answers to the Above Exercises
- (f−1)′(x)=13
- (g−1)′(x)=−1√1−x2
- (h−1)′(x)=1√1+x2
More References and links