How do I find dy/dx for sqrt(xy) = x - 2y using implicit differentiation?

In summary, the conversation was about finding the derivative of sqrt(xy) = x - 2y and simplifying it. The steps shown on Wolfram Alpha were followed and the final solution was y' = [- y + 2sqrt(xy)] / [x + 4sqrt(xy)]. The individual was unsure about the last step and sought clarification. After rearranging the terms, the correct solution was obtained.
  • #1
communitycoll
45
0

Homework Statement


Find dy / dx for sqrt(xy) = x - 2y.


Homework Equations



I don't know how to simplify

[(xy' + y) / 2sqrt(xy)] = (1 - 2y')

to

y' = [- y + 2sqrt(xy)] / [x + 4sqrt(xy)].


The Attempt at a Solution



I do everything Wolfram Alpha does here:

http://www.wolframalpha.com/input/?i=derivative+sqrt(xy)+=+x+-+2y

and at the end of the steps shown above, I want to multiply (1 - 2y') by 2sqrt(xy), to get

2sqrt(xy) - 4y'sqrt(xy)

although I don't think that's correct.

Help please.

I just realized all I d is solve for y' (didn't occur to me for some reason), and now need to know if I plug in the original equation into all of the y values (I think I remember seeing my prof. do this in class). Ayuda me por favor.
 
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  • #2
communitycoll said:
and at the end of the steps shown above, I want to multiply (1 - 2y') by 2sqrt(xy), to get

2sqrt(xy) - 4y'sqrt(xy)

although I don't think that's correct.

Yes, that's correct. Now rearrange the terms so that the ones with y' are on one side, and the other terms are on the other. Solve for y'.EDIT: It looks like you figured it out already.
 
  • #3
Thanks. I appreciate it.
 

FAQ: How do I find dy/dx for sqrt(xy) = x - 2y using implicit differentiation?

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function when the dependent variable cannot be expressed explicitly in terms of the independent variable. It is commonly used in multivariable calculus and is especially useful in solving equations involving curves that are not easily defined by a single function.

When should implicit differentiation be used?

Implicit differentiation should be used when the dependent variable cannot be easily isolated or expressed explicitly in terms of the independent variable. This often occurs when dealing with curves or equations that are not easily defined by a single function.

How is implicit differentiation different from regular differentiation?

Regular differentiation, also known as explicit differentiation, involves finding the derivative of a function where the dependent variable is expressed explicitly in terms of the independent variable. Implicit differentiation, on the other hand, involves finding the derivative of a function where the dependent variable is not explicitly expressed in terms of the independent variable, requiring the use of the chain rule and implicit differentiation rules.

What are the steps for using implicit differentiation?

The steps for using implicit differentiation are as follows:1. Differentiate both sides of the equation with respect to the independent variable.2. Use the chain rule to differentiate any terms involving the dependent variable.3. Gather all terms with the derivative of the dependent variable on one side of the equation.4. Solve for the derivative of the dependent variable.

What are some common mistakes to avoid when using implicit differentiation?

Some common mistakes to avoid when using implicit differentiation include:- Forgetting to use the chain rule when differentiating terms involving the dependent variable.- Mixing up the placement of the derivative symbol and the dependent variable.- Not simplifying the resulting equation after differentiating both sides.- Forgetting to use the product rule when differentiating products of functions.- Misapplying the power rule when differentiating terms raised to a power.

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