Partial and implicit differentiation

[hamza4best]

1. xy+cos(xy^2)=e^(x^2/y^3)

2. tan(x) + ln(x/y^2)=(x^2+y)/(y^3+x)

Differentiate w.r.t y

How do you know when to use partial differentiation and when to use implicit differentiation?
What I understood is that we apply implicit differentiation to equations and partial differentiation is applied to functions so in the above 2 cases we'll do implicit differentiation

Lastly as y is depending on x in the above equations so we wont treat x as a constant right?

 

[daon2]

1. xy+cos(xy^2)=e^(x^2/y^3)

2. tan(x) + ln(x/y^2)=(x^2+y)/(y^3+x)

Differentiate w.r.t y

How do you know when to use partial differentiation and when to use implicit differentiation?
What I understood is that we apply implicit differentiation to equations and partial differentiation is applied to functions so in the above 2 cases we'll do implicit differentiation

You can also partially differentiate implicitly! How you know which to perform should be clear from the question. If both subjects are being covered simultaneously, I wouldn't hesitate to clarify with the professor whether or not y is a function of x.

Lastly as y is depending on x in the above equations so we wont treat x as a constant right?

Edited: "Differentiate wrt y" in my mind says find d/dy of each term in your equation, treating x as constant. I would clarify what is being asked.

 

[hamza4best]

You can also partially differentiate implicitly! How you know which to perform should be clear from the question. If both subjects are being covered simultaneously, I wouldn't hesitate to clarify with the professor whether or not y is a function of x.


Right. You will need to treat y as a function of x, and be careful to use product/quotient rules when you see terms like xy, x^2/y^3, etc.[/QUOTE

We haven't learnt to partially differentiate implicitly up till now.The professor wanted us to figure that out by ourselves thats why its a bit confusing

 

[daon2]

I edited my post. After rereading your question I feel it is asking for you to differentiate both sides with respect to y. That is, treat each side as only functions of y.

 

[hamza4best]

I edited my post. After rereading your question I feel it is asking for you to differentiate both sides with respect to y. That is, treat each side as only functions of y.

then doesn't it mean that we're using partial differentiation to solve it then? as we're keeping x constant?
How do you figure that out which differentiation to use?

 

[daon2]

then doesn't it mean that we're using partial differentiation to solve it then? as we're keeping x constant?
How do you figure that out which differentiation to use?

You are not asked to find a derivative. That is, you are not asked to find "dy/dx" or "dx/dy." You have been handed an equation and asked to differentiate with respect to y. If you want to treat x as a function of y you may do so (but I see no indication that is wanted), with added work.

For example: say we want to differentiate xy = y^2 with respect to y. Assume that x is a function of y. You'll get dx/dy*y + x = 2y.

If x and y are truly unrelated quantities, then dx/dy = 0 simplifying your equation to x=2y, what you'd get for taking the partial derivative.

So again, without further confirmation that one of x or y is dependent on the other, it looks like only partial differentiation is requested.

 

[hamza4best]

You are not asked to find a derivative. That is, you are not asked to find "dy/dx" or "dx/dy." You have been handed an equation and asked to differentiate with respect to y. If you want to treat x as a function of y you may do so (but I see no indication that is wanted), with added work.

For example: say we want to differentiate xy = y^2 with respect to y. Assume that x is a function of y. You'll get dx/dy*y + x = 2y.

If x and y are truly unrelated quantities, then dx/dy = 0 simplifying your equation to x=2y, what you'd get for taking the partial derivative.

So again, without further confirmation that one of x or y is dependent on the other, it looks like only partial differentiation is requested.

I get what you mean now..But I thought when we're given an equation(not a function) then we always do implicit differentiation as x and y are mixed together and y is not explicit
And in partial differentiation we're always given a function like z=x^4+y^4

 

[daon2]

I get what you mean now..But I thought when we're given an equation(not a function) then we always do implicit differentiation as x and y are mixed together and y is not explicit
And in partial differentiation we're always given a function like z=x^4+y^4

The two are not mutually exclusive. If you are given a function of two independent variables: z=x^4+y^4, then there is no need for implicit differentiation to find, for example, \(\displaystyle \partial z/ \partial x\).

If instead you were told that z is a function of independent variables x and y and are asked to find \(\displaystyle \partial z/ \partial x\) for \(\displaystyle cos(z^2x)=xy^2\) then you will need to apply the partial derivative operator \(\displaystyle \partial/\partial x\) implicitly and solve for \(\displaystyle \partial z/\partial x\).