The rule for taking the derivative of the inverse of a function can be confusing. In this page I'll explain this topic in detail so you can leave without any doubt about it.
We have already used the rule for taking the derivative of a function. For example, we used it when calculating the derivative of inverse trig functions and also the derivative of ln(x). We'll go over these examples again here, and also deduce a general rule for any inverse function.So, what is the inverse of a function? Given a function f, its inverse, called f-1, is another function such that
This function is called the inverse because it returns to x when applied to f(x). That is, it inverts the effect of f. It is very important to note that not all functions have inverses.
Examples of inverse functions are the inverse trig functions. We have special names for these. For example, the inverse function of sin(x) is arcsin(x). In the page about inverse trig functions we saw how to calculate the derivative of these. And the idea is the same for any other inverse.
The essential idea is to apply the defining equation of an inverse, then use the chain rule.Let's denote the inverse function of f by g for a while, just to make the notation simpler. Because g is the inverse of f, we have that
Now we take the derivative of both sides of this equation. On the left side, we need to apply the chain rule.
Let's now introduce the variable
Replacing this we get
And this is the general rule for taking the derivative of an inverse function. Let's see some examples.
On the page derivative of ln(x) we calculated that derivative using geometric ideas. As much as I love that proof, I think that this is a case where an algebraic formula is better than the more intuitive idea.
Let's see how we can calculate this derivative using the general rule we just deduced.The function ln(x) is the inverse of the exponential ex. This is because, by definition
So, we can use the formula
To get an expression for the derivative of ln(x) using the formula, we need to calculate the derivative of f(x)
Replacing this into the formula, we get
On the page about inverse trig functions we calculated this one. Let's see how that calculation fits into the general idea we proved here. In this case we have
where g is the inverse of f. According to the formula
But we have that
Now, to have a pretty formula we need to write this as a function of f(x)=sin(x), because that is what we originally know. We can write
Then, the formula for the derivative of arcsin(x) is
If you want to see more examples of this, you can go to this page, where we deduce the formulas for the other inverse trig functions.
I think this gives you the general idea of how to take the derivative of an inverse. If you have a doubt, or want to ask about a specific problem, leave me a comment below.