Implicit Differentiation - 2nd derivative

In summary, the second derivative, d2y/dx2, for the equation xy + y^2 = 1 in terms of x and y is 2y(x+y)/(x+2y)^3. However, upon simplification, it becomes 2/(x+2y)^3 which is the correct answer. The mistake was not simplifying the equation enough.
  • #1
htoor9
17
0

Homework Statement



Find d2y/dx2 in terms of x and y for the following equation: xy + y^2 = 1 COMPLETELY SIMPLIFY

Homework Equations



dy/dx

The Attempt at a Solution



so i get -y/(x+2y) for the dy/dx. When I try to find the 2nd derivative and plug in dy/dx, I get 2y(x+y)/(x+2y)^3...the sheet says the correct answer is 2/(x+2y)^3. I don't understand I've done this problem 3 times and still don't get that answer!
 
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  • #2
I get your same answer. Perhaps your answer sheet is incorrect?
 
  • #3
Maybe...I don't know this is from a previous test. I feel like she should've fixed this by now?
 
  • #4
htoor9 said:
Maybe...I don't know this is from a previous test. I feel like she should've fixed this by now?

Oh my I am sorry I misled you like that. The answer is correct. Well both really. We just didn't simplify it enough!

y''=2y(x+y)/(x+2y)3 = 2(xy+y2)/(x+2y)3

But remember our starting equation? xy+y2= 1 :redface:
 

FAQ: Implicit Differentiation - 2nd derivative

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of an implicitly defined function, where the dependent variable is not explicitly expressed in terms of the independent variable. It is commonly used in cases where it is difficult or impossible to express the function explicitly.

Why do we need to find the 2nd derivative using implicit differentiation?

The 2nd derivative is important because it tells us about the rate of change of the rate of change of a function. In other words, it measures how the slope of a function is changing. This is useful in understanding the curvature and concavity of a curve, which can provide valuable information about the behavior of the function.

How is the 2nd derivative found using implicit differentiation?

To find the 2nd derivative using implicit differentiation, we first take the derivative of the original function to find the 1st derivative. Then, we take the derivative of the 1st derivative using the chain rule, where the dependent variable is treated as a function of the independent variable. This will give us the 2nd derivative in terms of both the independent and dependent variable.

What is the relationship between the 1st and 2nd derivatives using implicit differentiation?

The 1st derivative gives us information about the slope of a function, while the 2nd derivative tells us how that slope is changing. In other words, the 2nd derivative is the rate of change of the 1st derivative. This means that if the 2nd derivative is positive, the 1st derivative is increasing and the function is concave up, and if the 2nd derivative is negative, the 1st derivative is decreasing and the function is concave down.

What are some real-world applications of implicit differentiation and its 2nd derivative?

Implicit differentiation and its 2nd derivative are used in many fields of science and engineering, such as physics, chemistry, and economics. For example, it can be used to analyze the behavior of a chemical reaction, the motion of an object under the influence of forces, or the production and consumption of goods in an economy. It can also be used to optimize functions and solve optimization problems in real-world situations.

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