Differentiating related functions intro
Several verbal slips: says Y 'would just be F of T' (should be sqrt(F(T))), 'Y is a function of Y' (should be of T), and 'derivative of X with respect to 12' (should be with respect to T, equal to 12). The final answer of 2 is correct.
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X and Y are related by Y = sqrt(X), and X is itself a function of T, so Y is also a function of T. Using the chain rule, dY/dT = (dY/dX)(dX/dT) = (1/2)X^(-1/2)(dX/dT). With X = 9 and dX/dT = 12, this gives dY/dT = 2.
You'll be able to
- explain why Y depends on T when Y depends on X and X depends on T.
- write the chain rule as dY/dT = (dY/dX)(dX/dT).
- find dY/dT from a relating equation and given values.
Key words
- Differentiable function
- A function that has a derivative at every point we care about.
- Chain rule
- dY/dT = (dY/dX)(dX/dT) when Y depends on X and X depends on T.
- Power rule
- The derivative of X^n is n X^(n-1); for X^(1/2) it gives (1/2)X^(-1/2).
Transcript
- [Instructor] We are told the differentiable functions X and Y are related by the following equation, Y is equal to square root of X. And it's interesting, they're telling us that they are both differentiable functions, even X is a function, must be a function of something else.
Well, they tell us that the derivative of X with respect to T is 12 and they want us to find the derivative of Y with respect to T when X is equal to nine. So, let's just make sure we can understand this. So, they're telling us that both X and Y are functions.
Arguably they are both functions of T. Y is a function of X but then X is a function of T so Y could also be a function of T. One way to think about it is if X is equal to F of T, then Y is equal to the square root of X which would just be F of T. Another way to think about it if you took T as your input into your function F, you're going to produce X and then if you took that as your input into the square root function you are going to produce Y.
So, you could just view this as just one big box here, that Y is a function of Y but now let's actually answer their question. To tackle it we just have to apply the chain rule. The chain rule tells us that the derivative of Y with respect to T is going to be equal to the derivative of Y with respect to X times the derivative of X with respect to T.
So, let's apply it to this particular situation. We're gonna have the derivative of Y with respect to T is equal to the derivative of Y with respect to X. Well, what's that? Well, Y is equal to the principle root of X. You could also write this as Y is equal to X to the one half power.
We could just use the power rule. The derivative of Y with respect to X is one half X to the negative one half. So, let me write that down. One half X to the negative one half and then times the derivative of X with respect to T times the derivative of X with respect to T.
Well, let's see, we wanna find what we have here in orange, that's what the questions asks us. They tell us when X is equal to nine and the derivative of X with respect to T is equal to 12. So, we have all of the information necessary to solve for this. So, this is going to be equal to one half times nine to the negative one half, nine to the negative one half, times DX/DT, the derivative of X with respect to T is equal to 12, times 12.
So, let's see, nine to the one half would be three, nine to the negative one half would be 1/3, so this is 1/3, so this will all simplify to 1/2 times 1/3 is 1/6, so we could have a six in the denominator and then we are going to have a 12 in the numerator. So, 12/6, so the derivative of Y with respect to T when X is equal to nine and derivative of X with respect to 12 is two.
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