Section 4-6-3

Chapter 4
Differential Calculus and Its Uses

4.6 Derivatives of Functions Defined Implicitly

4.6.3 Derivatives of Inverse Functions

We have just seen two instances of the relationship between the derivatives of inverse functions — for the pair

y = \ln (x)

x = e^{y}

and for the pair

y = \sqrt{x}

x = y^{2} .

Since we will use this relationship again to determine other derivatives of inverses, it is worth our while to look at derivatives of inverse functions in general.

Suppose `f` and `g` are an inverse pair of functions: If `y=ftext[(]xtext[)]` then `x=gtext[(]ytext[)].` When we apply one after the other, we get

g (f (x)) = x .

Now we apply the Chain Rule to this equation to get

g^{'} (f (x)) f^{'} (x) = 1

f^{'} (x) = \frac{1}{g^{'} (f (x))} .

In differential notation, this becomes

\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} .

Thus, we see another instance (like the Chain Rule) in which differentials appear to behave just like numbers. In words, this formula says that derivatives of inverse functions are reciprocals of each other.

Example 2

Use the inverse function derivative formula, to give another derivation of the formula

\frac{d}{d x} \ln x = \frac{1}{x} .

Solution If `y=ln x`, then `x=e^y.` By the inverse derivative formula, we have

\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} = \frac{1}{e^{y}} = \frac{1}{x} .

Contents for Chapter 4

Chapter 4 Differential Calculus and Its Uses

4.6 Derivatives of Functions Defined Implicitly

Chapter 4
Differential Calculus and Its Uses