Derivatives of inverse functions: from equation | AP Calculus AB | Khan Academy
TLDRThe video script discusses the concept of inverse functions and their derivatives, using a specific function F and its inverse H as examples. It explains that if F and H are inverses, then the derivative of H with respect to a value is equal to 1 divided by the derivative of F with respect to the value that H maps to. The script applies this principle to find the derivative of H at -14, given that F of -2 is -14, and F'(x) is calculated using the power rule. The final result is 1/9, showcasing a unique application of calculus concepts.
Takeaways
- ๐ The function F(X) is defined as one half X to the third power plus three X minus four.
- ๐ H is the inverse function of F, denoted as H being the inverse of F.
- ๐ F(-2) is given as -14, which sets up the relationship between F and H.
- ๐ง Understanding the relationship between a function and its inverse is crucial for finding the derivative of the inverse function.
- ๐ The key property for inverse functions is that H'(X) = 1 / F'(H(X)), where H'(X) is the derivative of the inverse function H and F'(H(X)) is the derivative of the function F evaluated at H(X).
- ๐ The chain rule is fundamental in deriving the relationship between a function and its inverse.
- ๐ Since F and H are inverses, F(H(X)) = X and H(F(X)) = X by definition.
- ๐ค To find H'(-14), we first determine H(-14), which is -2 because of the inverse relationship.
- ๐ The derivative F'(X) is calculated using the power rule, resulting in (3/2)X^2 + 3.
- ๐งฎ Evaluating F'(-2) gives us 9, which is used to find H'(-14) as 1/9.
Q & A
What is the function F(X) defined as in the script?
-The function F(X) is defined as one half X to the third power, plus three X minus four.
What is the inverse function H in relation to F?
-H is the inverse function of F, meaning that applying H and then F (or vice versa) to any value will yield the original value.
What is the value of F at negative two?
-The value of F at negative two is negative 14.
What is the key property that relates the derivatives of a function and its inverse?
-The key property is that if F and H are inverses, then the derivative of H at X (H'(X)) is equal to one over the derivative of F at the inverse of X (1/F'(H(X))).
How does the chain rule relate to the derivatives of inverse functions?
-The chain rule is used to derive the property of the derivatives of inverse functions. By taking the derivative of both sides of the equation F(H(X)) = X and applying the chain rule, we can establish the relationship between the derivatives of F and H.
What is the value of H at negative 14, given that F of negative two is negative 14?
-Since F and H are inverse functions, H of negative 14 will yield the value that F of negative two maps to, which is negative two.
How is the derivative F'(X) calculated?
-The derivative F'(X) is calculated using the power rule. It is equal to three halves times X squared (3/2 * X^2) plus the derivative of three X (3).
What is the value of F'(negative two)?
-The value of F'(negative two) is calculated by substituting negative two into the derivative function F'(X), which results in three halves times the square of negative two plus three, equaling nine.
What is the value of H'(negative 14)?
-Since H'(negative 14) is the reciprocal of F'(H(negative 14)), and H(negative 14) is negative two, H'(negative 14) is the reciprocal of F'(negative two), which is one over nine.
Why is the problem in the script considered non-typical for a calculus class?
-The problem is considered non-typical because it involves finding the derivative of the inverse function at a specific point, which is not a common exercise in most calculus courses.
How does the script demonstrate the concept of inverse functions?
-The script demonstrates the concept of inverse functions by showing that applying the inverse function H to the result of F(negative two) yields the original input (negative two), and by explaining the relationship between the derivatives of the functions based on their inverse relationship.
Outlines
๐ Introduction to Function Inverses and Derivatives
This paragraph introduces the concept of function inverses and their relationship with derivatives. It presents a specific function F(X) and its inverse H, highlighting a known point where F(-2) = -14. The key property discussed is that if F and H are inverses, then the derivative of H with respect to X is equal to 1 divided by the derivative of F with respect to H(X). This relationship is derived from the chain rule and the fact that F and H being inverses implies F(H(X)) = X and H(F(X)) = X. The paragraph also touches on the difficulty of finding the inverse of a third-degree polynomial function and sets the stage for the problem-solving process to follow.
Mindmap
Keywords
๐กInverse Function
๐กDerivative
๐กPower Rule
๐กChain Rule
๐กFunction Evaluation
๐กRate of Change
๐กPolynomial Function
๐กDifferential Calculus
๐กRate of Change Reversal
๐กSlope
๐กMathematical Properties
Highlights
The concept of inverse functions and their relationship with derivatives is introduced.
F of X is defined as one half X to the third, plus three X minus four.
H is the inverse function of F, and F of negative two equals negative 14.
The main challenge is to find H prime of negative 14 using the properties of inverse functions.
A key property is that if F and H are inverses, then H prime of X equals one over F prime of H of X.
This property is derived from the chain rule in calculus.
F of H of X equals X, and H of F of X equals X, due to the nature of inverse functions.
Deriving both sides of the inverse function equation yields the key property of inverse function derivatives.
H prime of negative 14 is sought by utilizing the relationship between F and H.
Since F and H are inverses, H of negative 14 equals negative two.
F prime of X is calculated using the power rule, resulting in an expression involving X squared and a constant.
F prime of negative two is computed, leading to the evaluation of H prime of negative 14.
The final result is H prime of negative 14 equals one over nine, demonstrating the application of inverse function properties.
The problem showcases a non-typical calculus question involving inverse functions and their derivatives.
The discussion emphasizes the importance of understanding the chain rule and its application to inverse functions.
The transcript provides a comprehensive walkthrough of the problem, suitable for educational purposes.
The method presented can be applied to solve similar problems involving inverse functions and their derivatives.
The problem-solving approach is clear, logical, and well-structured, making it easy to follow.
Transcripts
- [Voiceover] Let F of X be equal to one half X to the third
plus three X minus four.
Let H be the inverse of F.
Notice that F of negative two is equal to negative 14.
And then they're asking us what is H prime of negative 14?
And if you're not familiar with the how functions
and their derivatives relate to their inverses
and the derivatives of the inverse, well this will
seem like a very hard thing to do.
Because if you're attempting to take the inverse of F
to figure out what H is
well, it's tough to find, to take
to figure out the inverse of a third degree
a third degree polynomial defined function like this.
So, the key
the key, I guess property to realize,
or the key truth to realize
if F and H are inverses
then H prime of X
H prime of X
is going to be equal to
is going to be equal to one over
F prime of H of X.
One over F prime of H of X.
And you could now use this
in order to figure out what H prime of
negative 14 is.
Now I know what some of you are thinking,
because it's exactly what I would be thinking
if someone just sprung this on me is
where does this come from?
And I would tell you, this comes straight out
of the chain rule.
We know that if a function and its inverse
we know that if we have a function and its inverse
that F of F of the inverse of our function.
So F of H of X.
F of H of X.
We know that this is going to be to X.
This literally, this is
comes out of them being each others inverses.
We could have also said H of F of X
will also be equal to X.
Remember, F is going to map
or H is going to map from some X to H of X.
And then F is going to map back to that original X.
That's what inverses do.
So that's because they are inverses.
This is by definition, this is what inverses
do to each other.
And then if you took the derivative of
both sides of this
what would you get?
Let me do that.
So if we took the derivative
of the both sides of this
D D X on the left hand side.
D D X on the right hand side.
I think you see where this is going.
You are essentially gonna get a version of that.
The left hand side used the chain rule.
You're going to get F prime of H of X.
F prime of H of X
times H prime of X
comes straight out of the chain rule
is equal to, is equal to the derivative of X
is just going to be equal to one.
And then you derive, you divide both sides by
F prime of H of X
and you get our original property there.
So now with that out of the way
let's just actually apply this.
So, we want to evaluate H prime of 14.
Or sorry, H prime of negative 14.
Is going to be equal to
one over F prime of
H of negative 14.
H of negative 14.
Now had they given us H of negative 14.
But they didn't give it to us explicitly,
we have to remember that F and H are inverses
of each other.
So F of negative two is negative 14.
Well, H is gonna go from the other way around.
If you input negative 14 into H
you're going to get negative two.
So H of negative 14
well, this is going to be equal to
negative two.
Once again, they are inverses of each other.
So H of negative 14 is equal to
negative negative two.
And once again, I just swapped these two around.
That's what the inverse function will do.
If you're wrapping from if F goes from
negative two to negative 14
H is going to go from negative 14
back to negative two.
So now we want to evaluate F prime of negative two.
Well, let's figure out what F prime of X is.
So, F prime of X is equal to
remember the power rule, so three times one half
is three halves
times X to the three minus one power
which is the second power.
Plus the derivative of three X
with respect to X which that's just going to be three.
And you could do that, it's just the power rule.
But this was X to the first power,
one times three,
X to the zero power,
but X to the zero is just one
so you're just left with three.
And derivative of a concept that's just gonna be zero.
So that's F prime of X.
So F prime of, F prime of negative two
is going to be three halves
times negative two squared is four, positive four.
So plus three.
So, this is going to be equal to
two times three plus three.
So, six plus three is equal to nine.
So this denominator here is going to be equal to nine.
So this whole thing is equal to one over nine.
So this involved, this was something you're not
going to see every day.
This isn't that typical problem in your calculus class.
But it's interesting.
Browse More Related Video
5.0 / 5 (0 votes)
Thanks for rating: