Solving a Calculus Problem with Implicit Differentiation

In summary, the conversation is about implicit differentiation and a problem involving it. The person is trying to find dy/dx and is asking for clarification on where to apply the implicit differentiation rule. The expert explains that when differentiating x^2, you simply get 2x, but when differentiating y^2, you must use the chain rule and multiply by dy/dx. The person then asks for confirmation on their work, which the expert deems as correct.
  • #1
Jason03
161
0
Heres another problem I was working on...

http://img141.imageshack.us/img141/2318/calc2qg4.jpg

Im trying to find dy/dx using implicit differentiation...my algebra is a bit rusty...but I am trying to make sure I am on the right track...
 
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  • #2
Implicit diff of x^2 is not 2xy', it's just 2x. Everything else is fine.
 
  • #3
where did I do that in the problem?
 
  • #4
Not quite;

When you are differentiating x2 with respect to x, for example, you simply get 2x. You do not multiply 2x by dy/dx.

However, when you (implicitly) differentiate y2 with respect to x (where y is a function of x) you get

[tex]
2y \frac{dy}{dx}
[/tex]

Here you do multiply by dy/dx; the reason comes from the chain rule.
 
  • #5
Jason03 said:
where did I do that in the problem?

Third line, first expression on the left hand side (referring to your attached image)
 
  • #6
like this?

http://img375.imageshack.us/img375/5230/calc3eh3.jpg
 
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  • #7
does the work above look ok?
 
  • #8
Jason03 said:
does the work above look ok?

Looks fine to me
 

1. What is implicit differentiation?

Implicit differentiation is a technique used in calculus to find the derivative of an equation that is not explicitly written in terms of one variable. This is commonly used when solving equations involving multiple variables or when the dependent variable cannot be easily isolated.

2. When should I use implicit differentiation?

Implicit differentiation is most commonly used when solving problems involving curves or surfaces that cannot be easily expressed as functions of a single variable. It is also useful when dealing with equations that involve transcendental functions such as trigonometric or exponential functions.

3. How do I solve a calculus problem using implicit differentiation?

To solve a problem using implicit differentiation, follow these steps:

  1. Differentiate both sides of the equation with respect to the independent variable.
  2. Use the chain rule to differentiate any functions that involve multiple variables.
  3. Isolate the derivative on one side of the equation.
  4. Substitute known values or variables to find the value of the derivative at a specific point.

4. What are the 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 functions with multiple variables.
  • Mistakes in algebraic simplification when isolating the derivative.
  • Not substituting the values of known variables correctly.
  • Forgetting to include the derivative notation, such as d/dx, when writing the final answer.

5. How can I check if my answer is correct when using implicit differentiation?

To check if your answer is correct when using implicit differentiation, you can:

  • Verify that the original equation and the derivative equation are both satisfied by the same values of the independent and dependent variables.
  • Plot the original equation and the derivative equation on a graph to see if they have the same slope at various points.
  • Use an online graphing calculator or a computer algebra system to verify your answer.

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