Recall that the most important property of an inverse function is that it *undoes* the action of a function, and vice versa. In function notation it looks like this:

$$f(f^{-1}(x)) = f^{-1}(f(x)) = x$$

It might help to think of a function as an "**operator**" which **operates** on the variable **x**. If the function **f** operates on **x**, then operation by the inverse function **f ^{-1}** immediately afterward simply

It works the other way too. Operation by **f** on the result of operating on **x** by **f ^{-1}** just undoes the first action. Inverse operations undo each other.

We recognize the importance of inverse operations when we work with trigonometric and exponential functions. We can't find an angle in a right triangle from the lengths of its sides without an inverse trig. function (sin** ^{-1}**(x), cos

**Inverse functions undo the action of each other**

The exponent "-1" in the definition of an inverse function does not mean what it usually means. It does not mean "take the reciprocal" as it usually does. When associated with a function name like $f^{-1}(x),$ it denotes the inverse function, which is *not* the reciprocal of $f(x).$

Graphically, a function and its inverse are mirror images across the line $y = x.$ Take the example plotted below. The inverse of $f(x) = x^2$ is the square root function, $f^{-1}(x) = \sqrt{x}.$ Notice that for the root function, we have to restrict ourselves to the upper arm of the sideways parabola, otherwise it would be double-valued. That's very common with inverse functions (see Inverse Trig. Functions).

The blowup on the right gives you a closer view. Now the way that reflection across the line $y = x$ works is kind of cool. If a point $(x, y)$ exists on $f(x),$ then the point $(y, x)$ is its mirror image on $f^{-1}(x).$ For example, $(2, 4)$ is on the graph of $f(x)$ below, and $(4, 2)$ is its mirror image on $f^{-1}(x).$

This mirror-image property will help us a lot as we take derivatives of inverse functions.

**y = x**. If (x, y) is on f(x), then (y, x) is its mirror-image point across y = x, and the slope of f(x) at x is the reciprocal of the slope of f^{-1}(x) at y.

Now let's take a generic function, **f(x)**, and its inverse **f ^{-1}(x)**. We begin with the statement

$$f(f^{-1}(x)) = x$$

Now we differentiate both sides with respect to **x**,

$$\frac{d}{dx} f(f^{-1}(x)) = \frac{d}{dx}x$$

remembering that we need to apply the chain rule to the composition of functions on the left:

$$f'(f^{-1}(x))[f^{-1}(x)]' = 1$$

If we divide to isolate the derivative of the inverse on the left, we finally get:

$$[f^{-1}(x)]' = \frac{1}{f'(f^{-1}(x))}$$

$$\frac{d}{dx} \left[ f^{-1}(x) \right] = \frac{1}{f'(f^{-1}(x))}$$

The derivative of an inverse function, **f ^{-1}(x)** can be found without directly taking the derivative, if we know the function,

Finding the derivatives of the main inverse trig functions (sine, cosine, tangent) is pretty much the same, but we'll work through them all here just for drill

We're looking for

$$\frac{d}{dx} sin^{-1}(x)$$

If we let

$$y = sin^{-1}(x)$$

then we can apply **f(x) = sin(x)** to both sides to get:

$$sin(y) = x$$

Now if we take the derivative of each side with respect to **x** (**d/dx**), remembering that we have to use implicit differentiation on the left, we get:

$$cos(y) \frac{dy}{dx} = \frac{dx}{dx} = 1$$

On the right is just = 1, so we have the derivative we're looking for:

$$\frac{dy}{dx} = \frac{1}{cos(y)}$$

... but we want it in terms of **x**. That's easy because we know that **sin(y) = x**. We can build a triangle that reflects that fact: the sine of angle **y** is **x**:

$$$$

To get the bottom side, just use the Pythagorean theorem. Now we can replace **cos(y)** with an algebraic expression containing **x**:

$$\frac{d}{dx} sin^{-1}(x) = \frac{1}{\sqrt{1 - x^2}}$$

$$\frac{d}{dx} tan^{-1}(x)$$

First let **y = tan ^{-1}(x)**, then apply

$$tan(y) = x$$

Now taking the derivative of both sides gives:

$$sec^2(y) \frac{dy}{dx} = \frac{dx}{dx} = 1$$

Then we divide by **sec ^{2}(x)** to get

$$\frac{dy}{dx} = \frac{1}{sec^2(y)}$$

Now using **tan(y) = x**, we can construct another triangle:

... and from that we find

$$\frac{d}{dx} tan^{-1}(x) = \frac{1}{1 + x^2}$$

This derivative is calculated in much the same way. We'll skip the details for this one; you should try it on your own. The result is:

$$\frac{d}{dx} cos^{-1}(x) = \frac{-1}{\sqrt{1 - x^2}}$$

You could use the same method to find derivatives of the inverse cosecant, secant and cotangent functions, too. Try it!

Show that the derivative of $sin^{-1}(x)$, as obtained above, is equivalent to the derivative found using $\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$

**Solution**

$$f(x) = sin(x) \:\: \& \:\: f^{-1}(x) = sin^{-1}(x) $$

We need the derivative of the function

$$f'(x) = cos(x)$$

Then it's just a matter of plugging the inverse in to cos(x):

$$f'(f^{-1}(x)) = \frac{1}{cos(sin^{-1}(x))}$$

Now it's a little difficult to convert this into the form we found in example 1, but if we plot the two, the result is in the graph on the right.

*I've shifted the purple curve upward by 0.1 so you can see that the curves would perfectly overlap otherwise.*

This graph is another version of the $f(x) = x^2$ , $f^{-1}(x) = \sqrt{x}$ graph above. Point (2, 4) is shown on **f(x)**, and its mirror image across **y = x**, (4, 2) is shown on **f ^{-1}(x)**.

Let's say we want to know the derivative (slope) of the inverse function at x = 4, but we don't actually know the inverse function (I know we know it here, but pretend we don't). Turns out we don't really need to know f^{-1}(x).

If (4, 2) is a point on f^{-1}(x), then (2, 4) is the point on f(x) at which f(x) has the reciprocal slope.

The green lines are tangent to the functions at those points. They have reciprocal slope. That means that the slope of the inverse at x = 4 can be found by taking the derivative of **f(x)** at x = 2.

We saw above that if points **(a, b)** and **(c, d)** lie on **f(x)**, then points **(b, a)** and **(d, c)** will lie on **f ^{-1}(x)**. We can use those two points to calculate the slopes of the segments connecting those lines. On the function, that slope is

$$m_f = \frac{d - b}{c - a}$$

and on the inverse it is.

$$m_{f^{-1}} = \frac{c - a}{d - b}$$

It's easy to recognize that these are reciprocals of each other:

$$m_f = \frac{1}{m_{f^{-1}}}$$

locations **a**, **b**, **c** and **d** are completely arbitrary, so we could make the distance between our two points as close to zero as we want, as we would in taking a derivative, so the relationship between derivatives is:

$$\frac{d}{dx}f(x) = \frac{1}{\frac{d}{dx} f^{-1}(x)}$$

1. |
Let $f(x) = 2x^5 + x^3 + 1$. Find $\frac{d}{dx}f^{-1}(x)$ at x = 4. ## SolutionThe slope of the inverse function at x = 4 will be the reciprocal of the slope of the $$ \begin{align} f'(x) &= 10x^4 + 3x^2 \\[4pt] f'(1) &= 10 + 3 = 13 \end{align}$$ So the slope of f'(x) at x = 4 is $\frac{1}{13}$ |

2. |
Given that $f(x) = x^3 + 7x + 2$ and $f(1) = 10$, calculate the value of $f^{-1}(10)$. ## SolutionIf the point (1, 10) is on the graph of f(x), then the point (10, 1) is on the graph of $f^{-1}(x).$ So the value is 1. |

3. |
Given that $f(x) = 7x^3 + (ln(x))^3$ and $f(1) = 7$, calculate the value of $f^{-1}(7)$. ## SolutionIf the point (1, 7) is on the graph of f(x), then the point (7, 1) is on the graph of $f^{-1}(x).$ So the value is 1. |

4. |
Find the equation of the line tangent to the inverse of $f(x) = x^5 + 2x^3 + x - 4$ at the point (-4, 0). ## SolutionIf the point (-4, 0) is on $f^{-1}(x),$ then the point (0, -4) is on f(x). That's easy to confirm. Now if we find the slope of f(x) at x = 0, then the slope of $f^{-1}(x)$ at x = -4 is the reciprocal of that slope. $$ \begin{align} f'(x) &= 5x^4 + 6x^2 + 1 \\[4pt] f'(0) &= 1 \end{align}$$ So the slope of the inverse is also 1. |

5. |
Take a look at the table below, showing values of a function $f(x)$ and its derivative f'(x). Use the table to find $(f^{-1})'(1)$ and $(f^{-1})'(-3)$. You will likely encounter problems like this on the AP calculus exam. ## SolutionHere we use $$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$ $(f^{-1})'(1)$: If $f^{-1}(1) = y,$ then we also know that $f(y) = 1.$ From the table, we see that $f(-1) = 1,$ so $f^{-1}(1) = -1,$. Using the table again, we just search for $f'(-1),$ which is $\frac{-1}{5}.$ So our derivative is -5. $(f^{-1})'(-3)$: If $f^{-1}(3) = y,$ then we also know that $f(y) = -3.$ From the table, we see that $f(4) = -3,$ so $f^{-1}(-3) = 4,$. Using the table again, we just search for $f'(4),$ which is $-2$ So our derivative is $\frac{-1}{2}.$ |

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