Derivatives of inverse functions | Advanced derivatives | AP Calculus AB | Khan Academy

Summary

The video explains the relationship between a function and its inverse, particularly focusing on their derivatives. Using the chain rule, the instructor derives a formula connecting the derivative of a function to the derivative of its inverse. By applying this concept to specific functions like \( e^x \) and its inverse, the natural log, the video demonstrates how this relationship holds true. This technique offers a useful method for calculating the derivative of an inverse function. The video encourages viewers to review inverse functions and apply calculus concepts for deeper understanding.

Takeaways

• 🧮 Inverse functions have the property that g(f(x)) = x, where g(x) is the inverse of f(x).
• 🔄 If f maps x to a value f(x), then the inverse function g takes f(x) back to x.
• 📝 The chain rule can be applied to find the derivative of inverse functions.
• 📉 Taking the derivative of both sides of g(f(x)) = x results in a useful relationship between the derivatives of a function and its inverse.
• ⚙️ Using the chain rule, we find that f'(x) = 1 / g'(f(x)).
• 🔍 This equation provides a link between the derivative of a function and the derivative of its inverse.
• 📚 For example, when f(x) = e^x, its inverse is g(x) = ln(x).
• 💡 The derivative of e^x is e^x, and the derivative of ln(x) is 1/x, supporting the derived relationship.
• 🔄 The formula holds true for other functions as well, verifying the relationship between inverse functions and their derivatives.
• 🧠 This concept helps in understanding and computing the derivative of an inverse function more efficiently.

Q & A

What is the relationship between two functions that are inverses of each other?

If f(x) and g(x) are inverse functions, then applying g to f(x) (or f inverse of f(x)) results in x, meaning g(f(x)) = x.

How do inverse functions map values?

An inverse function g maps the output of f(x) back to the original input x. If f(x) takes x to f(x), then g(f(x)) brings it back to x.

What rule is applied to find the derivative of inverse functions?

The chain rule is applied to find the derivative of inverse functions, leading to a relationship between the derivatives of a function and its inverse.

How do you express the relationship between the derivatives of inverse functions?

The derivative of f(x), f'(x), is equal to 1 divided by the derivative of g at f(x), or f'(x) = 1 / g'(f(x)).

Why is the relationship between the derivatives of inverse functions useful?

This relationship allows you to calculate the derivative of an inverse function if you know the derivative of the original function.

What happens when you differentiate the equation g(f(x)) = x using the chain rule?

Using the chain rule, the derivative of g(f(x)) = x results in g'(f(x)) * f'(x) = 1, which helps derive the formula f'(x) = 1 / g'(f(x)).

How can you verify the derivative relationship for exponential and logarithmic functions?

For f(x) = e^x and g(x) = ln(x), f'(x) = e^x and g'(x) = 1/x. Substituting into the formula f'(x) = 1 / g'(f(x)) confirms the result.

What is the derivative of e^x, and why is it significant?

The derivative of e^x is e^x itself, which is a unique and important result in calculus because the function equals its own derivative.

What is the derivative of the natural logarithm function?

The derivative of the natural logarithm function ln(x) is 1/x.

Can you use the derivative relationship to find the derivative of inverse functions without directly differentiating them?

Yes, using the relationship f'(x) = 1 / g'(f(x)), you can deduce the derivative of an inverse function by knowing the derivative of the original function.

Outlines

00:00

🔄 Introduction to Inverse Functions and Their Relationship

The instructor begins by explaining the concept of inverse functions. If f(x) is a function, g(x) is its inverse, meaning that applying f(x) and then g(x) returns the original value, x. The instructor reviews this concept, explaining that f(g(x)) or g(f(x)) equals x. He encourages viewers to review inverse functions if they are unfamiliar, referencing Khan Academy for further understanding.

🔗 Applying the Chain Rule to Inverse Functions

The discussion moves to applying calculus to inverse functions using the chain rule. The instructor differentiates both sides of the equation g(f(x)) = x. The chain rule gives the derivative of g with respect to f(x), g'(f(x)), multiplied by the derivative of f(x), f'(x). The derivative of x with respect to x is simply 1. This sets up a key relationship between the derivatives of a function and its inverse.

📊 Deriving the Key Relationship Between a Function and Its Inverse

The instructor simplifies the derived equation by dividing both sides by g'(f(x)), yielding the formula: f'(x) = 1 / g'(f(x)). This formula shows a relationship between the derivative of a function and its inverse. This result allows one to deduce the derivative of the inverse function given the derivative of the original function.

📐 Example with Exponential and Logarithmic Functions

The instructor illustrates the derived formula using the functions f(x) = e^x and g(x) = ln(x). He walks through how the inverse of e^x is ln(x) by solving for the inverse algebraically. This serves as a review of inverse functions for those who need it, with the reminder to check Khan Academy if necessary.

🔄 Validating the Formula for f(x) = e^x and g(x) = ln(x)

The instructor tests whether the formula f'(x) = 1 / g'(f(x)) holds for the exponential function f(x) = e^x and its inverse, g(x) = ln(x). He computes the derivatives: f'(x) = e^x and g'(x) = 1/x. Substituting these into the formula, he confirms that it holds true, as 1 / (1 / e^x) simplifies to e^x.

🔄 Generalization to Any Inverse Functions

The instructor generalizes the result, explaining that for any pair of inverse functions, g'(x) = 1 / f'(g(x)) will always hold true. This relationship can help find the derivative of an inverse function using the original function's derivative. The session concludes by emphasizing how neat and useful this result is in understanding derivatives of inverse functions.

Keywords

💡Inverse Function

An inverse function reverses the effect of the original function, meaning if you apply a function to a value and then apply its inverse, you will return to the original value. In the video, f(x) and g(x) are introduced as inverse functions, where g(x) is the inverse of f(x) and vice versa. The inverse function concept is key to understanding how the derivative of a function relates to its inverse.

💡Derivative

A derivative represents the rate of change of a function with respect to its variable. In the video, the derivative of f(x) and its inverse, g(x), are used to explore the relationship between a function and its inverse in calculus. The chain rule is applied to take derivatives of these functions, revealing how the derivatives of inverse functions interact.

💡Chain Rule

The chain rule is a formula in calculus for finding the derivative of a composite function. In the video, it is applied to the composition of f(x) and g(x) (where g is the inverse of f) to differentiate the combined function g(f(x)), yielding an important relationship between their derivatives.

💡g(f(x)) = x

This equation expresses a key property of inverse functions: applying a function and its inverse successively results in the original input. In the video, this concept is used to set up the calculation for the derivative of g(f(x)) using the chain rule, which leads to a significant relationship between f’(x) and g’(f(x)).

💡f'(x) = 1/g'(f(x))

This equation shows the relationship between the derivative of a function and the derivative of its inverse. Derived through the chain rule in the video, this formula reveals that knowing the derivative of a function can help determine the derivative of its inverse. This is central to the video's message about the interplay between functions and their inverses.

💡Exponential Function (e^x)

The exponential function e^x is a standard example in calculus because its derivative is uniquely equal to itself. In the video, f(x) is set as e^x, and the process of finding the derivative of its inverse, the natural logarithm, is demonstrated. This function is used to verify the derived formula f'(x) = 1/g'(f(x)).

💡Natural Logarithm (ln(x))

The natural logarithm function, ln(x), is the inverse of the exponential function e^x. In the video, g(x) is defined as ln(x) when f(x) is e^x, and the derivatives of both are explored. The relationship between ln(x) and e^x helps to illustrate the connection between a function and the derivative of its inverse.

💡f'(x) = e^x

This is the derivative of the exponential function e^x, which is central to the example used in the video. The video uses this derivative in conjunction with the inverse function, ln(x), to demonstrate that f'(x) is consistent with the formula 1/g'(f(x)).

💡g'(x) = 1/x

This is the derivative of the natural logarithm ln(x), which is used in the video to confirm the relationship between the derivatives of a function and its inverse. The video shows that using this derivative for g(x) verifies the formula f'(x) = 1/g'(f(x)) when f(x) is e^x.

💡Function Composition

Function composition involves applying one function to the result of another, expressed as g(f(x)). In the video, this concept is essential for understanding the relationship between f(x) and g(x) as inverse functions. The chain rule is applied to this composition to explore how their derivatives relate.

Highlights

Introduction to inverse functions: understanding f(x) and its inverse g(x).

Inverse functions property: g(f(x)) or f inverse of f(x) equals x.

Explanation of how functions f and g map values and return to the original input.

Review of inverse functions: f maps x to f(x) and g takes f(x) back to x.

Application of the chain rule to differentiate the equation g(f(x)) = x.

Derivation result: f prime of x is equal to 1 divided by g prime of f(x).

Explanation of the relationship between the derivative of a function and its inverse.

Example with f(x) = e^x and its inverse g(x) = ln(x).

Review of the natural logarithm function as the inverse of the exponential function.

Calculation of f prime of x for f(x) = e^x, showing f prime equals e^x.

Derivative of natural log function: g prime of x = 1/x.

Verification of the derived result: e^x equals 1 over (1/e^x), confirming the theory.

Concept of using derivatives of inverse functions to calculate each other.

Further exploration: g prime of x is equal to 1 over f prime of g(x), due to their inverse relationship.

Application of the result to general inverse functions for calculating derivatives.

Transcripts

- [Instructor] So let's say I have two functions

that are the inverse of each other.

So I have f of x,

and then I also have g of x,

which is equal to the inverse of f of x.

And f of x would be the inverse of g of x as well.

If the notion of an inverse function

is completely unfamiliar to you,

I encourage you to review inverse functions on Khan Academy.

Now, one of the properties of inverse functions

are that if I were to take g of f of x,

g of f of x, or I could say the f inverse of f of x,

that this is just going to be equal to x.

And it comes straight out of what

an inverse of a function is.

If this is x right over here,

the function f would map to some value

f of x.

So that's f of x right over there.

And then the function g, or f inverse,

if you input f of x into it, it would take you back,

it would take you back to x.

So that would be f inverse,

or we're saying g is the same thing as f inverse.

So all of that so far is a review of inverse functions,

but now we're going to apply a little bit of calculus to it,

using the chain rule.

And we're gonna get a pretty interesting result.

What I want to do is take the derivative

of both sides of this equation right over here.

So let's apply the derivative operator,

d/dx on the left-hand side,

d/dx on the right-hand side.

And what are we going to get?

Well, on the left-hand side, we would apply the chain rule.

So this is going to be the derivative of g

with respect to f of x.

So that's going to be g prime of

f of x,

g prime of f of x,

times the derivative of f of x with respect to x,

so times

f prime of x.

And then that is going to be equal to what?

Well, the derivative with respect to x of x,

that's just equal to one.

And this is where we get our interesting result.

All we did so far is we used

something we knew about inverse functions,

and we'd use the chain rule

to take the derivative of the left-hand side.

But if you divide both sides by g prime of f of x,

what are you going to get?

You're going to get a relationship

between the derivative of a function

and the derivative of its inverse.

So you get f prime of x

is going to be equal to

one over all of this business,

one over g prime

of

f of x,

g prime of f of x.

And this is really neat because if you know something

about the derivative of a function,

you can then start to figure out things

about the derivative of its inverse.

And we can actually see this is true

with some classic functions.

So let's say that f of x

is equal to

e to the x,

and so g of x

would be equal to the inverse of f.

So f inverse,

which is, what's the inverse of e to the x?

Well, one way to think about it is,

if you have y is equal to e to the x,

if you want the inverse, you can swap the variables

and then solve for y again.

So you'd get x is equal to e to the y.

You take the natural log of both sides,

you get natural log of x is equal to y.

So the inverse of e to the x is natural log of x.

And once again, that's all review of inverse functions.

All right, if that's unfamiliar, review it on Khan Academy.

So g of x is going to be equal to

the natural log of x.

Now, let's see if this holds true for these two functions.

Well, what is f prime of x going to be?

Well, this one of those amazing results in calculus.

One of these neat things about the number e is that

the derivative of e to the x is e to the x.

And in other videos, we also saw that the derivative

of the natural log of x is one over x.

So let's see if this holds out.

So we should get a result, f prime of x,

e to the x

should be equal to

one over

g prime of f of x.

So g prime of f of x,

so g prime is one

over our f of x,

and f of x is e to the x,

one over e to the x.

Is this indeed true?

Yes, it is.

One over, one over e to the x

is just going to be e to the x.

So it all checks out.

And you could do the other way

because these are inverses of each other.

You could say g prime of x is going

to be equal to one over f prime of g of x

because they're inverses of each other.

And actually, what's really neat about this,

is that you could actually use this to get a sense of what

the derivative of an inverse function is even going to be.