Reviewing Inverses and Functions | Monotonic Functions in an Interval |

Finding the Derivative of an Inverse Function |

Around the time you’re studying **Exponential and Logarithmic Differentiation**, you’ll probably learn how to get the **derivative of an inverse function**. The reason we might need to do this is it might be much easier to get the derivative of an inverse function than to take the inverse of a function and then take the derivative! In fact, the derivations of the exponential and log derivatives were a direct result of differentiating inverse functions.

## Reviewing Inverses of Functions

We learned about **inverse functions** here in the **Inverses of Functions** section. You get the inverse of a function if you switch the $ x$ and $ y$ and solve for the “new $ y$”. A function has an inverse function if it is **one-to-one **(or** invertible**), which means it passes both vertical and horizontal line tests. A function that has an inverse or is one-to-one is strictly **monotonic **(either increasing or decreasing) for its entire domain.

## Monotonic Functions in an Interval

We can determine if a function is **monotonic** in an interval (and therefore **has an inverse** in that interval) if the derivative of that function is either greater than **0** (increasing) or less than **0** (decreasing) for that entire interval.

Let’s first do some problems where we use the derivative to find out if a function has is strictly **monotonic** (has a strictly increasing or decreasing derivative) on its entire domain:

## Finding the Derivative of an Inverse Function

The derivative of an inverse function is found using the equation below; it looks really intimidating, but if you take it step-by-step, it’s not that bad. Note that $ f\left( {g\left( x \right)} \right)$ means a **composite function **(which we learned about here in the** Advanced Functions: Compositions, Even and Odd, and Extrema **section), which means that we take the inside function, $ g\left( x \right)$, and put that in everywhere there’s an “$ x$” in the outside function, $ f\left( x \right)$.

What this says is if we have a function and want to find the **derivative of the inverse of the function** at a certain point “$ x$”, first find the “$ y$” for the particular “$ x$” in the original function. Then, use this value as the “$ x$” in the derivative of this function and finally take the reciprocal of this number. This gives the derivative of the inverse of the original function at this point. Another way to explain this is “the derivative of $ f\left( x \right)$ at a point $ (a, b)$ is the reciprocal of the derivative of $ {{f}^{{-1}}}\left( x \right)$ at point $ (b, a)$”. I know this makes no sense at this point, but we’ll do problems below.

Remember that **we must first check that the function is monotonic in the given interval**, to make sure the function is one-to-one (has an inverse).

Here are some **Derivative of the Inverse** problems; notice how they may be presented in different ways. Some teachers may have you solve these using **Implicit Differentiation**, so I’m including that method, too. (You may also want to review how to find roots of polynomials in the **Graphing and Finding Roots of Polynomial Functions** section.)

On to **Antiderivatives and Indefinite Integration, including Trig Integration** – you are ready!