How do I find v' for functions v=x+y using implicit differentiation?

In summary, the conversation is about finding the derivative of a function with respect to a given variable and using the chain rule to do so. The process involves differentiating the function's definition and applying the chain rule to find the derivative. The conversation also touches on finding the derivative of a function involving multiple variables and using the product rule and chain rule together to do so.
  • #1
daster
Say we have two functions of x, v and y, such that v=x+y. How can I find v'?
 
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  • #2
Why doncha differentiate its definition wrt to "x"??

Daniel.

PS.Tell me what u get.
 
  • #3
Since v and y are functions of x, I presume by " v' " you mean "the derivative of v with respect to x", i.e. dv/dx.

Use the chain rule: v= x+ y so dv/dx= dx/dx+ dy/dx= 1+ dy/dx. What dy/dx is depends, of course, on what function y is of x.
 
  • #4
Oh, so it's only dy/dx? Cause I remember my book doing something like (dy/dx)(dv/dx) or something. Thanks HallsofIvy. :smile:

Another question. How do I find d(e^u dy/dx)/du, where u and y are functions of x?
My book says:

[tex]e^u \frac{dy}{dx} + e^u \frac{d^2y}{dx^2} \cdot \frac{dx}{du}[/tex]

I understand this is the product rule, but where'd the dx/du come from?
 
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  • #5
It comes from the chain rule again:

[tex]\frac{d}{du} \frac{dy}{dx}=\frac{d^2y}{dx^2}\frac{dx}{du}[/tex]
 
  • #6
Dank je. :smile:
 
  • #7
Graag gedaan. :biggrin:
 

1. What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is given implicitly, meaning the dependent and independent variables are not explicitly defined in the equation. In other words, the equation is not in the form of y = f(x).

2. How is implicit differentiation different from explicit differentiation?

Explicit differentiation is used to find the derivative of a function that is given explicitly, meaning the dependent and independent variables are clearly defined in the equation. Implicit differentiation, on the other hand, is used for equations that are not explicitly defined.

3. What is the process of implicit differentiation?

The process of implicit differentiation involves taking the derivative of both sides of the equation with respect to the independent variable. Then, use the rules of differentiation to solve for the derivative of the dependent variable. The result will be in terms of both the dependent and independent variables.

4. When is implicit differentiation used?

Implicit differentiation is used when the equation cannot be solved for the dependent variable explicitly and when the derivative of the dependent variable is needed.

5. What are some real-world applications of implicit differentiation?

Implicit differentiation is commonly used in physics, engineering, and economics to solve problems involving rates of change and optimization. It can also be used in curve fitting and data analysis.

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