What is Implicit differentiation: Definition and 279 Discussions

In mathematics, an implicit equation is a relation of the form R(x1,…, xn) = 0, where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x2 + y2 − 1 = 0.
An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. For example, the equation x2 + y2 − 1 = 0 of the unit circle defines y as an implicit function of x if –1 ≤ x ≤ 1, and one restricts y to nonnegative values.
The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function.

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  1. G

    Implicit Differentiation Check

    Homework Statement just a check, in Implicit Differentiation if you have let's say (x2+y2)2 would you get 2(x2+y2)(2x+2y(dy/dx)) or would it go out of the whole function in the chain rule and be 2(x2+y2)(2x+2y)(dy/dx) much appreciated.
  2. G

    Implicit Differentiation: Finding dy/dx for a Given Equation

    Homework Statement Implicit Differentiation: I Was given the equation find dy/dx: (3x3y2 + 7x) (x2y3 + 3xy)- The Attempt at a Solution Ok, i know i have to use the product rule on top, and on bottom and the quotient rule for the fraction so... if i set s = 3x3 t = y2 u = (3x3y2 +...
  3. L

    Use implicit differentiation to find dy/dx [Answer check]

    Use implicit differentiation to find \frac{dy}{dx} for xy^{2} – yx^{2} = 3xy i've answered the question but i think I'm doing it wrong any help is appreciated! x(2y)\frac{dy}{dx} – y(2x) = 3xy 2xy \frac{dy}{dx} – 2yx = 3xy 2xy\frac{dy}{dx} = 5xy \frac{dy}{dx} = \frac{5xy}{2xy}...
  4. J

    Implicit differentiation solving a function

    Im trying to solve this function, that is part of a larger equation: \frac{d}{dx} xe^y do I need to get rid of the x term so the y term is dy/dx?
  5. J

    Solving a Calculus Problem with Implicit Differentiation

    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...
  6. T

    Query regarding implicit differentiation

    Homework Statement Use implicit diff to find partial x and partial y if x² + y² + z² = 3xyz The Attempt at a Solution firstly with partial x... 2x + 2z(∂z/∂x) = 3yz + 3xy(∂z/dx) 2x - 3yz = 3xy(∂z/dx) - 2z(∂z/∂x) 2x - 3yz = (3xy - 2z)(∂z/dx) (2x - 3yz)/(3xy - 2z) = (∂z/dx)...
  7. R

    When is the derivative of a function defined using implicit differentiation?

    Hello! I need some help here please for ppl who are familiar with implicit differentiation. Use implicit differentiation to find dy/dx, in each case say where it is defined; a) y^5 +x^2 y^3 = 1+xy b) y= \frac{x^{3/2}\sqrt{7x^2 +1}}{sin(x) e^{3x^2 + 2x}}, x \neq n\pi n \in Z...
  8. S

    Easy Implicit Differentiation Question

    Homework Statement y^2 = 3x^3 + 2x and y must be positive. Find the normal component of acceleration when: x=3m \dot x = 5ms^{ - 1} \ddot x = 5ms^{ - 2} 2. The attempt at a solution Well my approach would be to differentiate it implicitly twice and solve for {\ddot y}. Then I have...
  9. D

    How can I solve this implicit differentiation problem?

    so I have a implicit diffentiation problem and was wondering if someone could help me out.. I need to figure out how to get dy/dx=0 so eg if i had dy/dx = 4xy+2x/5y^2 and you want to write this in terms of y, how is this done? is there a trick?
  10. J

    What is Implicit Differentiation and How is it Related to the Chain Rule?

    Could someone please explain to me how implicit differentiation is an application of the chain rule? It would be much appreciated. By the way, if it helps, I'm a junior in high school. Thanks.
  11. S

    Find dy/dx by implicit differentiation: 6x^2+8xy+y^2=6

    Homework Statement Find dy/dx by implicit differentiation: 6x^2+8xy+y^2=6 Homework Equations n/a The Attempt at a Solution I'm using y'=(dy/dx) I found the derivative of the above problem. 12x+8xy'+10y=0 (I used the product rule to find the derivative of 8xy') 12x+8xy'+10y=0...
  12. R

    How do I use implicit differentiation to solve these two questions?

    Im having trouble solving these two questions. I don't know where to start so I can't give an attempt at either of them. Please tell me how to do the full question if you can cause i can't check back until morning then i have to go to an exam. http://www.users.on.net/~rohanlal/one.jpg...
  13. R

    Implicit Differentiation: Solving for dy/dx in (x^2-y^2)^2=(x+y)^3

    (x^2-y^2)^2=(x+y)^3 I tried to use the chain rule on both sides but it didn't work because y needs to have the chain rule used on it explicitly and if i differentiate y explicitly then use the chain rule on everything i would be finding the 2nd derivative. So how do i differentiate this?
  14. F

    Implicit differentiation to find the slope

    Homework Statement (x - h)^2 + (y - k)^2 = r^2 where h,k and r are constants The Attempt at a Solution \begin{array}{l} \frac{d}{{dx}}\left[ {(x - h)^2 + (y - k)^2 } \right] = \frac{d}{{dx}}r^2 \\ 2(x - h) + 2\frac{{dy}}{{dx}}(y - k) = 0 \\ \Rightarrow \frac{{dy}}{{dx}}...
  15. M

    Implicit differentiation dy/dx=xy^2

    Homework Statement If dy/dx=xy^2 and x=1 when y=1, then y= (A) x^2 (B) -2/(x^2 -3) (C) x^2 + 3 (D) 2/(x^2 +1) (E) (x^2 -3)/2 Homework Equations The Attempt at a Solution dy/dx=xy^2 dy=xy^2dx dy/y^2=x dx ∫dy/y^2=∫x dx -1/y = x^2/2 + C y=-2/x^2 + C 1=-2/1^2 + C C=3...
  16. U

    Implicit differentiation? implicit integration?

    We have implicit differentiation: i.e. x^2 + y^2 = 7 -> 2x +2y(dy/dx) = 0. and solve for dy/dx gives you the derivative of y with respect to x However, is there not implicit integration? for terms linear in y, i.e. x^2 + y = 7 -> X^3/3 +int(y) = 7x + K, and solve for int(y) to...
  17. B

    Implicit Differentiation of Cylinder NOT given radius?

    Implicit Differentiation of Cylinder NOT given radius? Homework Statement Question: Digging in his backyard, Dennis accidentally breaks a pipe attached to his water sprinkling system. water bubbles up at a rate of 1 cm^3/s, forming a circular pond of depth 0.5cm in his yard. How quickly is...
  18. S

    Implicit differentiation and chain rule-

    The question and my workings are attached:
  19. H

    Implicit Differentiation [SOLVED]

    [SOLVED] Implicit Differentiation -4x^2+3xy+4y^3=-328 This is at the point (3,-4) and i am trying to find m Homework Equations The Attempt at a Solution here is my work -4x^2+3xy+4y^3=-328 -8x+3xy'+3y+12y^2y'=0 -8x+3y=-3xy'-12y^2y' -8x+3y=[-3x-12y^2]y'...
  20. H

    What is the error in this implicit differentiation problem at the point (1,3)?

    [SOLVED] Implicit Differentiation [b]-4x^2+3xy+4y^3=-328 This is at the point (1,3) Homework Equations The Attempt at a Solution here is my work -4x^2+3xy+4y^3=-328 -8x+3xy'+3y+12y^2y'=0 -8x+3y=-3xy'-12y^2y' -8x+3y=[-3x-12y^2]y' y'=(-8x+3y)/(-3x-12y^2) plug in x=1,y=3...
  21. O

    Implicit differentiation question

    Find y' = dy/dx for x3 + y3 = 4 Okay, now what's really confusing me is that for the y3 is that you need to use the chain rule for it. When you do, the answer is 3y2(dy/dx). How does that actually work? And if anyone can give me any good advice on any good guidelines on how to properly...
  22. B

    How to Solve Implicit Differentiation Problems with Tan(xy)=xy?

    okay, so i have the problem tan(xy)=xy and i am told to implicitly differentiate in terms of x and y. when i do, i keep getting an indefinite answer. i need help because i know that's not right.
  23. B

    Implicit differentiation of one-parameter family

    Homework Statement Use implicit differentiation to show that the one parameter family f(x, y)=c satisfies the differential equation dy/dx = -f_{x}/f_{y} , where f_{x}=\frac{\partial f}{\partial x} and f_{y}=\frac{\partial f}{\partial y} .Homework EquationsThe Attempt at a Solution Well, my...
  24. A

    How Do You Solve Second Derivatives Using Implicit Differentiation?

    Hi, I need some help with these question, and would appreciate the help. Homework Statement Part 1 Use implicit differentiation to find y'' if 2xy = y^2 Simplify and Leaev in terms of x and y. Part 2 Use implicit differentiation to find y '' if xy + y^3 = 1 Simplify your answer and...
  25. R

    Implicit differentiation wrt time

    Homework Statement "A lawn sprinkler is constructed in such a way that d\theta/dt is constant, where theta ranges between 45 degrees and 135 degrees. The distance the water travels horizontally is x=v^2sin(2\theta)/32 where v is the speed of the water. Find dx/dt and explain why this lawn...
  26. N

    A few implicit differentiation problems

    1.Use implicit differentiation to find the equation of the tangent line to the curve xy^3+xy=16 at the point (8,1) . The equation of this tangent line can be written in the form y=mx+b 2.For the equation given below, evaluate y^1 at the point(1,-1) . (6x-y)^4+2y^3=2399. 3.Find the...
  27. B

    Implicit Differentiation with Trig

    Homework Statement Find the slope of the tangent line to x tan y = y - 1 when y = pi/4 Homework Equations The Attempt at a Solution I can't seem to get the derivative. Here's what I do. First I used the product rule the left side of the equation and got sec^2 x dy/dx + tan...
  28. R

    Hard? implicit differentiation

    Homework Statement Find d^2y/dx^2 in terms of x and y. x^2y^2-2x=3 Homework Equations property rule, chain rule, quotient rule, The Attempt at a Solution I can do this the long way, but there must be a shorter solution. Can I simplify it? I've found dy/dx=(-xy^2 +2x)/(2x^2y)...
  29. K

    Determine (dy/dx) using implicit differentiation

    Determine (dy/dx) using implicit differentiation. cos(X^2Y^2) = x I'm really confused what to do now..i think the next steps are: d/dx [cos(X^2*Y^2)] = d/dx [x] = -sin(X^2*Y^2)* ((X^2*2Y dy/dx) + (Y^2*2X)) = 1 = -2YX^2 sin(X^2*Y^2) dy/dx + -2XY^2sin(X^2*Y^2) = 1 = -2YX^2 sin(X^2*Y^2)...
  30. M

    How do you differentiate (x-y) using implicit differentiation?

    I looked through my notes and couldn't figure out how to differentiate (x - y) using implicit differentiation. Could someone help with that and I should be able to work out the rest of my question :)
  31. M

    Find dy/dx by implicit differentiation

    Question: Find dy/dx by implicit differentiation 4x^2 + 3xy - y^2 = 6 Attempt: Ok, just a forewarning that I suck at differentiations, limits, what-not, so.. Following my textbook, it says I should Differentiate both sides of the equation So... (d/dx)(4x^2 + 3xy - y^2) = (d/dx)(6)...
  32. M

    Implicit Differentiation Problem

    Homework Statement x^{}2(x-y)^{}2=x^{}2-y^{}2 Homework Equations The Attempt at a Solution I can get this far: x[2(x-y)(1-dy/dx)]+2x(x-y)=2x-2ydy/dx Any small hint as to where to go from here would be much appreciated.
  33. L

    Implicit Differentiation is so confusing

    Hello there! Please help me relieve my confusion. Thanks! For \frac{d}{dx}[y^{3}] , why do you need to use the chain rule on this equation? Basically, the chain rule is used on almost every function right? It is just that we do not see the dx/dx since it equals one, for example...
  34. R

    Implicit Differentiation: Finding Higher Order Derivatives and Exact Solutions

    Homework Statement Hi everyone, name is Ryan. This is my first post here, seems like a very beneficial forum. I look forward to being helped and helping others. Anyway I'm trying to teach myself Implicit Differentiation but there doesn't seem like much useful resources online and I don't quite...
  35. homology

    How do I solve an implicit differentiation problem with two different answers?

    Hi guys, I have a question on an implicit differentiation problem. I get two different answers depending on how I do it and the answers are different (not just looking, but different). The problem is [tex] \frac{x+3}{y}-4x-y^2=0 [\tex]. One option is to just differentiate as it stands and you...
  36. C

    Implicit Differentiation: How to Find the Derivative of an Unknown Function?

    Homework Statement I just got started on this, and am not grasping the WHOLE idea. 1.xy=25 The answer says -y/x 2.x^2+3xy+y^2=15 And this says -y^2/x^2 Homework Equations 1. dy/dx(xy)= dy/dx(25) 1=0 ? 2.dy/dx x^2+3xy+y^2= dy/dx 15 2x+3+y(dy/dx) =0 The...
  37. S

    Implicit Differentiation problem

    Homework Statement Find dy/dx by implicit differentiation when it is known that y^2 + xsiny = 4 Homework Equations The Attempt at a Solution 2y dy/dt + xcosy dy/dt + siny = 0 2y dy/dt + xcosy dy/dt = -siny dy/dt + dy/dt = -siny/2y/xcosy I'm sure I'm doing it wrong so I...
  38. M

    Find dy/dx for Implicit Differentiation of cos(xy^2)-(x^3+y)/(x+1)=sec(x)sin(y)

    assistance asap is needed ! Using implicit differentiation, find dy/dx given that: cos(xy^2) - (x^3+y) / (x+1) = sec(x)sin(y) i am horrid at these, i came up with a few lines, please check them -sin(xy^2)[(y)^2+(2y)(dy/dx)(x)] - [ {(3x^2)(x+1)-(x^3+y)} / (x+1)^2] = sec(x)tan(x)sin(y) +...
  39. G

    Problems with implicit differentiation

    In a problem where I need to use implicit diff. to find the slope of a line such as: y^2 + x^2 = 9 2y (dy/dx) + 2x = 0 dy/dx = -y/x Where does the (dy/dx) after the 2y (in the second part) come from? I've already differentiated y^2, x^2, and 9. Why isn't it just 2y +...
  40. S

    Implicit Differentiation Problem

    Homework Statement If (1+x^2)(y^2)=1-x^2, show that (\frac{d}{{dx}})^2=\frac{1-y^4}{{1-x^4}} 2. The attempt at a solution http://img294.imageshack.us/img294/7133/calcqn1qd8.gif I have got to this point and tried to simplify the problem with no success ... Have I made an error in my...
  41. H

    Implicit differentiation homework

    Question: Find the co ords of the turning points of y^3 + 3xy^2 - x^3 = 3 Attempt: (differentiate w.r.t. x) d/dx(y^3 + 3xy^2 - x^3) = d3/dx 3y^2(dy/dx) + 3(2xy(dy/dx) + y^2) - 3x^2 = 0 (divide through by 3) y^2(dy/dx) + 2xy(dy/dx) = x^2 - y^2 (take dy/dx as a comon factor) dy/dx(y^2 +...
  42. T

    Is My Implicit Differentiation Method Correct?

    Homework Statement http://img224.imageshack.us/img224/2459/untitledow9.jpg Homework Equations The Attempt at a Solution See above picture. I'm just curious to see if my method is correct and how exactly would I go about simplifying the answer if indeed it is correct. Thanks...
  43. V

    How Fast Is the Shadow's Height Changing as the Dog Approaches the Wall?

    Homework Statement A spotlight on the ground shines on a wall 14m away. If a dog, 0.5m tall, runs from the spotlight towars the building at a speed of 1 m/s, how fast is the height of the animal's shadow on the building decreasing when the dog is 5 meters from the building? Wrt = with...
  44. A

    Another implicit differentiation

    e^x^y = x +y ok i know i am suppost to use the chain rule and the product rule so x+y is 1 +1 if u find the derivatives, but e^x^2 is confusing me, what is u and what is n i think u= e^x^2 and n= y is that possible for n to equal y, this problem is confusing
  45. N

    Finding y'' for Implicitly Differentiated x^4+y^4=1

    find y'' (double prime) for x^4+y^4=1
  46. S

    Question Involving implicit Differentiation

    Hi! I have a problem here that's been bugging me. I was wondering if anyone can give insight into where I'm going wrong implicit differentiation problem 1) (x^2+y^2)/(x+y)=xy-2 find derivitive (dy/dx) at point (-1, -1) I know the basic premise. I used the quotient rule to find the...
  47. L

    Using Implicit Differentiation

    Alright I have the question: Find dy/dx by implicit differentiatin ysin(x^2) = xsin(y^2) Basically you jus take the derivative of both sides and solve for dy/dx, but I was unsure whether or not my differentation was right. If someone could just get me started in the right direction for...
  48. D

    Another implicit differentiation question

    ok so i did this problem but i wasnt sure if this is correct. y = 8x + 5cos(xy) + 7 dy/dx = 8 + 5(-sin xy)(x dy/dx + y) dy/dx = 8 - (5x) (sin xy) dy/dx - (5y) (sin xy) (5x) (sin xy) dy/dx + dy/dx = 8 - (5y) (sin xy) [(5x) (sin xy) + 1] dy/dx = 8 - (5y) (sin xy) dy/dx = [8 - (5y) (sin...
  49. D

    Implicit differentiation question

    I have this question in which I know I probably have to use implicit differentiation but I have no idea how to do this can someone give me a hint to get started. all the implicit differentiation problems I have done only have a combination of x and y but this one has x, y and t. find dy/dx...
  50. E

    Help with Partial Derivatives and Implicit Differentiation

    The problem is as follows: Cartesian and polar coordinates are related by the formulas x = r\cos\theta y = r\sin\theta Determine \frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}. Differentiate the...
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