Implicit differentiation find dy/dx and d2y/dx2

In summary, the conversation is about finding the first and second derivatives of a given equation and confirming the correctness of the answer. The first derivative is correctly found to be (3-2y)/(2x+2y) and the second derivative is calculated to be (-12x+2x^2y+9+4yx-14y-4y^2)/((x+y)(2x+2y)^2). The person asking the question is unsure if the second derivative can be simplified further. The answer is yes, and it was found using the initial equation provided.
  • #1
mattmannmf
172
0
2xy=3x-y^2

find dy/dx and d2y/dx2

I just want to make sure my answer is right and simplified

I got dy/dx= (3-2y)/ (2x+2y)

Now d2y/dx2 took some time but this is what i got:

(-12x+2x2y+9+4yx-14y-4y^2) / ((x+y)(2x+2y)^2)
 
Physics news on Phys.org
  • #2
Your dy/dx is correct. I didn't work out the second derivative, but as long as you took the derivative of dy/dx using the quotient rule and implicit differentiation, replacing dy/dx in that expression with what you found, and then simplifying correctly, then you did it right.
 
  • #3
yeah, i did that must i just wasn't to sure with my algebra (thats usually the step i would mess up). thanks though!
 
  • #4
I don't know if you can reduce this to a more simpler form but this is simplest i got: y"=(3-2y)(3+8x+6y)/8(x+y)^3
 
  • #5
was this from the equation i have up there or did u do this from the beginning of the problem?
 
  • #6
the answer is yes and yes.
 

FAQ: Implicit differentiation find dy/dx and d2y/dx2

1.

What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of a function that is not explicitly in the form of y=f(x). It involves differentiating both sides of an equation with respect to x and using the chain rule to find the derivative of the dependent variable, y.

2.

How do you find dy/dx using implicit differentiation?

To find dy/dx using implicit differentiation, you first differentiate both sides of the given equation with respect to x. Then, isolate the term that contains dy/dx and solve for it. This will give you the derivative of the dependent variable, y, with respect to the independent variable, x.

3.

What is d2y/dx2 and how is it related to implicit differentiation?

d2y/dx2, also known as the second derivative, is the derivative of the derivative. It represents the rate of change of the first derivative, dy/dx. In implicit differentiation, d2y/dx2 is found by differentiating the first derivative, dy/dx, with respect to x using the chain rule.

4.

Can implicit differentiation be used to find higher order derivatives?

Yes, implicit differentiation can be used to find higher order derivatives, such as d3y/dx3 and d4y/dx4, by differentiating the previous derivative with respect to x using the chain rule. However, the process can become more complex and time-consuming as the order of the derivative increases.

5.

What are some common mistakes made when using implicit differentiation to find derivatives?

One common mistake is not applying the chain rule correctly when differentiating the dependent variable, y. It is important to identify and differentiate the outer function first, and then multiply by the derivative of the inner function. Another mistake is not using proper notation, such as using d/dx instead of dy/dx. It is also important to double-check the final answer and make sure all terms are simplified and in the correct form.

Similar threads

Solve this problem that involves implicit differentiation
Replies
15
Views
1K
Exploring an Alternative Approach to Implicit Differentiation
Replies
2
Views
1K
Implicit differentiation: why apply the Chain Rule?
Replies
3
Views
1K
Find the second derivative of the relation; ##x^2+y^4=10##
Replies
24
Views
2K
Question on Two Differentiation Techniques
Replies
25
Views
1K
First, second and third deriative of ln (tan x )
Replies
1
Views
2K
I need some help with implicit differentiaiton.
Replies
5
Views
1K
Implicit Differentiation: How Do I Solve \dfrac{x^2}{x+y}=y^2+8?
Replies
22
Views
2K
Implicit Differentiation Question
Replies
2
Views
1K
Write the tangent lines to the curve
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top