Implicit differentiation: two answers resulted

In summary, there are two different ways to solve the problem of finding dy/dx of \frac{x}{x+y}-\frac{y}{x}=4. The first method involves using the quotient rule, while the second method involves simplifying the equation before taking the derivative. After solving for dy/dx in both methods, the resulting equations are different, with the first method yielding \frac{dy}{dx}=y/x and the second method yielding \frac{dy}{dx}=-\frac{6x+5y}{2y+5x}. This difference may be due to a mistake in the second line of the second method, where it should be -y^2 instead of -2y^2. Further
  • #1
123
1
Could anyone explain that I got two different answers for this question: find [tex]dy/dx[/tex] of [tex]\frac{x}{x+y}-\frac{y}{x}=4[/tex].

1. using quotient rule:
[tex]\frac{x+y-(1+dy/dx)x}{(x+y)^{2}}-\frac{x\frac{dy}{dx}-y}{x^{2}}=0[/tex]
[tex]\frac{y}{(x+y)^{2}}-\frac{x}{(x+y)^{2}}\frac{dy}{dx}+\frac{y}{x^{2}}-\frac{1}{x}\frac{dy}{dx}=0[/tex]
[tex](\frac{x}{(x+y)^{2}}+\frac{1}{x})\frac{dy}{dx}=\frac{y}{(x+y)^{2}}+\frac{y}{x^{2}}[/tex]
[tex]x(\frac{1}{(x+y)^{2}}+\frac{1}{x^{2}})\frac{dy}{dx}=y(\frac{1}{(x+y)^{2}}+\frac{1}{x^{2}})[/tex]
[tex]\frac{dy}{dx}=y/x[/tex]

2. simplify using common denominator before taking derivative:
[tex]\frac{x^{2}}{x^{2}+xy}-\frac{xy+y^{2}}{x^{2}+xy}=4[/tex]
[tex]x^{2}-xy-y^{2}=4x^{2}+4xy[/tex]
[tex]-y^{2}=3x^{2}+5xy[/tex]
[tex]-2y\frac{dy}{dx}=6x+5y+5x\frac{dy}{dx}[/tex]
[tex]\frac{dy}{dx}=-\frac{6x+5y}{2y+5x}[/tex]
 
Last edited:
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  • #2
[tex]
x^{2}-xy-y^{2}=4x^{2}+4xy
[/tex]
[tex]
y^{2}=3x^{2}+3xy
[/tex]

The second line does not follow from the first.
 
  • #3
Sorry, it's suppose to be "5xy"
 
  • #4
and it's suppose to be -y^2. Now I corrected it.
 
  • #5
Help, anyone? i still don't get why.
 

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not expressed explicitly in terms of one variable. It involves differentiating both sides of an equation with respect to the variable of interest.

Why do two answers result from implicit differentiation?

Two answers can result from implicit differentiation because the technique requires differentiating both sides of an equation, which can lead to multiple solutions or answers.

How can implicit differentiation be applied in real-world situations?

Implicit differentiation can be applied in real-world situations to solve problems involving rates of change, such as in physics, economics, and engineering. It can also be used to find the slope of a curve at a specific point.

What are the limitations of using implicit differentiation?

One limitation of implicit differentiation is that it can only be applied to functions that can be expressed implicitly in terms of one variable. It also requires a strong understanding of calculus and algebraic manipulation.

Can implicit differentiation be used to find higher order derivatives?

Yes, implicit differentiation can be used to find higher order derivatives by differentiating both sides of the equation multiple times. This can be useful in solving more complex problems and understanding the behavior of a function.

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