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# Worked example: Differentiating related functions

In this video, we tackle a fascinating problem involving related functions. We'll explore how to use implicit differentiation to find the derivatives of these functions with respect to a shared variable. By the end, we'll have a clear understanding of how these functions interact and change together.

## Want to join the conversation?

• hi i am very confused :(
• Don't be discouraged, calculus can be very difficult for many people. It would be helpful if you could be specific about what confuses you, making sure you include the time on the video which corresponds to the place where you start to get confused. Good luck!
• Oh i've just used pythagorian identity to find cos(x), is anything wrong with it?
• No, there is nothing wrong with that. You can use any method you want as long it's correct.
• How did d/dt become d/dx and d/dy? I thought the chain rule only worked for functions, so how can you derive taking the derivative of something?
• The chain rule does only work for functions! Luckily, both x and y in this case are functions of t, so we can use the chain rule.
• Instead of doing the algebra at the end like Sal did at , couldn't you get straight to 5 because we know that y=x and the derivative of x with respect to t is equal to the derivative of y with respect to t. We know that the derivative of x with respect to t is equal to 5, so the derivative of Y with respect to t is also 5. Is this thinking wrong?
• You're not wrong; however, it was important for Sal to do the question as he did for the sake of his explanation, jumping the gun would probably confuse a lot of learners.
(1 vote)
• At , isn't the value of x both π/4 and 3π/4 making the final answers both 5 and -5?
• Well, not quite.
Notice that the question mentioned that x must be between 0 and pi/2, if the domain was different, your answer would work.
(1 vote)
• at , i instead took d/dx (sin x + cos y)=d/dx √2, rahther than taking them based on d/dt, and then i got dy/dx=cos x/sin y.

since dy/dt=(dy/dx)*(dx/dt), after substitution i got the same result 5, it's still correct to do it this way, right? Cheers!
(1 vote)
• Problem is, you can't differentiate w.r.t x anymore. In differentiation, we can only differentiate w.r.t the independent variable. In normal circumstances, x is independent and y is dependent, which is why we can do d/dx but not d/dy. Here, both x and y are dependent on t. Hence, you can only differentiate w.r.t t here, as it is the only independent variable.
• I feel like I'm getting the hang of this, but there's one particular class of questions that I'm completely lost on: take something like 3x^2 = xy. I can do almost any other question by following the kind of steps in this video, but when I try to apply it to this one, for some reason I get the wrong answer. The problem seems to happen with the "xy" part specifically - I always assume each should be converted to the derivative at the same step that "3x^2" is, so as follows:

1) 3x^2 * dx/dt = (x * dx/dt) * (y * dy/dt)

2) 6x * dx/dt = (1 * dx/dt) * (1 * dy/dt)

...because if I'm not mistaken, 1 is the derivative of a single variable without a multiplier or power. But the explanation for the answer tells me this is incorrect and that I should instead be taking the base values of x and y, as in the following:

2) 6x * dx/dt = (x * dx/dt) * (y * dy/dt)

This appears intuitively wrong to me because it's inconsistent; why would I differentiate the x on one side but not the x and y on the other? Nevertheless, it's apparently considered to be the right way to do it, because the other way gives incorrect answers. Can someone please tell me what I'm doing wrong?
(1 vote)
• Both your answer and the one given seem to be wrong. Here's why.

First off, a bit of notation: d/dt[3x^2] is what you should be writing. It's definitely not the same thing as 3x^2 * dx/dt.

Left hand side: d/dt[3x^2] = d[3x^2]/dx * dx/dt (chain rule) = 6x * dx / dt.
Right hand side: d/dt[xy] = dx / dt * y + dy / dt * x (product rule).

6x * dx / dt = (x * dy / dt) + (y * dx / dt).

Video on chain rule and product rule: https://www.youtube.com/watch?v=YG15m2VwSjA.
• Why does sinx change to cosx in and later cos x change to sinx
(1 vote)
• Because they are taking the derivative of sinx which is cosx and the derivative of cosx which is -sinx. As you can see from your time stamp, d/dx is removed from the equation. Which happens when you take the derivative.