What is Implicit: Definition and 535 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. 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...
  2. 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)...
  3. 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)...
  4. 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 :)
  5. A

    Answer Quadratic Implicit Function: Horizontal/Vertical Tangent Line

    Homework Statement a. Find dy/dx given that x^{2}=y^{2}-4x=7y=15 b. under what conditions on x and/or y is the tangent line to this curve horizontal? vertical? 2. The attempt at a solution I did solve the first question by simply using implicit fuction. 2x+2y*y'-4+ty*y'=0...
  6. 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)...
  7. 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.
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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) +...
  14. 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 +...
  15. 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...
  16. 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 +...
  17. 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...
  18. 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...
  19. P

    Explicit vs implicit time dependence in Lagrangian mechanics

    I'm trying to understand something that's coming from my Marion & Thornton (4th edition 1995 on p. 264 in a section titled "Conservation Theorems Revisited"). The topic is conservation of energy and introduction of the Hamiltonian from Lagrange's equations. We're told that the Lagrangian...
  20. P

    Optimizing Implicit Functions: Best Fit for Coupled ODE Solutions

    Is it possible to make a least squares fit with a function given implicitly, because the equation isn't solveable analyticly? Because I had the coupled ODE, \ddot{x} = \omega^2x + 2\omega\dot{y} - C\,\frac{\dot{x}}{\dot{r}} \ddot{y} = \omega^2y - 2\omega\dot{x} - C\,\frac{\dot{y}}{\dot{r}}...
  21. 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
  22. A

    Find dy/dx by implicit differetiation HELP IMMEDIATLY

    of 4 cos x sin y =1 here's what i did d/dx 4 cosx sin y)= 1 d/dx 4sinx cosy=0 y'= 4 sinx cosy but the answer is tan x tan y
  23. N

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

    find y'' (double prime) for x^4+y^4=1
  24. L

    Solve Implicit Function Question: Derivatives, Jacobi Matrix, Diff.

    Hi! I've got a question about implicit functions. I have to solve a system f(x,y,z)=0 in the neighbourhood of (1,1,1). I have a problem computing the derivative of an implicit function (x,y)=g(z), whose existence is given by the implicit function theorem when applied to the given function...
  25. N

    Mathematica Is there an implicit age limit for mathematical productivity?

    Some mathematicians note that their intellectual powers (at least where mathematics is concerned) seem to diminish with age, for instance Hardy. Was this griping a mere excuse for their lack of talent to begin with? Other prodigies appeared to have retained their mathematical fecundity into...
  26. P

    Dynamical systems:From explicit to implicit equations

    Hi! This is my first post...I've a little question about a mathematical issue I found in the passage from explicit to implicit equations of a dynamical system. How to demonstrate that?? http://pixhost.eu/show_big.php?/share/2007-01-19/doi.jpg Thanks to all
  27. C

    Implicit function on functions composed of itself

    Suppose F(x, y) is C1. F(0, 0) = 0. What conditions on F will guarantee that the equation F(F(x, y), y) = 0 can be solved for y as a C1 function of x near (0, 0) ? would it simply be dF/dy not equal 0 ?
  28. C

    Question about implicit function theorem

    if z was solved in terms of x, y, then when we differenciate d (dz/dx) / dx, are we treating z as a constant or still a function of x, y ?
  29. 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...
  30. S

    Implicit Derivation: Show That y' Satisfies Equation x sin(xy)+2x²=0

    the equation x sin (xy) +2x² defines y implicitly as a function of x. assuming the derivative y' exists, show that it satisfies the equation y'x² cos (xy) +xy cos(xy)+sin (xy)+4x = 0. Help needed please. I found the derivative of the first equation is: sin xy + xy cos xy +4x. It's close...
  31. R

    Proof of the implicit and inverse function theorems

    Today I revised my knowledge from multivariable calculus and I found that I couldn't remember the proofs of these two theorems. Then I looked in Rudin, and everything was clear. Except one thing, which probably made me forgot the proofs. There are two weird functions in these two proofs...
  32. 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...
  33. S

    Finding Tangent Lines Through the Origin to Implicit Differentiation

    I can't get the problem. can anyone help me please. -Find equations for two lines thorugh the origin that are tangent to the curve x^2 - 4x +y^2 + 3 = 0. I found dy/dx=(-x+2)/y and put thta into the point slope equation, and then filled in (0,0) for the point, but couldn't get an...
  34. 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...
  35. 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...
  36. 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...
  37. E

    Using implicit differentiation to differentiate log_a (x)

    Could someone please make sure I'm doing this right. I want to find the derivative of the logarithm to the base a of x, using implicit differentiation. Let y = \log_{a} x a^y = x \frac{d}{dx} (a^y) = 1 (implicit differentiation) \frac{d}{dx} (e^{\ln a})^y = 1 \frac{d}{dx} (e^{(\ln a)y}) = 1...
  38. C

    Rank condition in the Implicit Mapping Theorm

    Hi there. I've recently come across the Implicit Mapping Theorm in my studies and noticed that there is a condition that the rank of the image must be the maximum possible. I'm not directly seeing why this condition is needed, so I was wondering if anyone could provide me with an example of why...
  39. N

    Implicit Differentiation - Second Derivative

    Were assigned questions regarding implicit differentiation and the second derivative but did not receive a formal lesson so I need some explanations. Example: Find the second derivative x^3 + y^3 = 1 I found this solution on the internet and the answer matches the one in the textbook...
  40. A

    ThanksWhere Can I Find Software to Plot Implicit Functions?

    Can anybody give me links to download software that can plot implicit functions like x^2 + xy =9 etc. I have searched the net but all that i have found are shareware demos that expire after a period.I am sure there are freeware plotters of this kind but can't find any.
  41. B

    Implicit Differentiation in Multivariable Calculus

    I need to compute the partials of z with respect to x and y of: xy + z + 3xz^5 = 4 at (1,0). I already showed that the equation is solvable for z as a function of (x,y) near (1,0,1) with the special implicit function theorem, but that's the easy part. Could someone explain to me how to begin...
  42. V

    Implicit Differentiation Problem

    Consider the curve given by X^2+4y^2=7+3xy a) show that dy/dx=3y-2x/8y-3x b) show that there is a point P with x-cooridnate 3 at which the line tangent to the curve at P is horizontal. Find the y-cooridnate of P. c)find the value of d^2y/dx^2 at the point P found in part (b). Does the curve...
  43. M

    Find an explicit or implicit solutions to the differential equation, what now?

    Hello everyone, yet another obscure problem on web work. No examples like this in the book nor did the professor go over it so i was wondering if someone can let me know what exactly they are wanting me to do! Find an explicit or implicit solutions to the differential equation...
  44. Q

    Finding the Derivative of a Function with Implicit Differentiation

    Assume that y is a function of x . Find y' = dy/dx for (x^3+y^3)^20 when i solved this i got y'= (20(x^3+y^3)^19 * 3x^2)/(-3y^2) is this correct or am i missing something?
  45. T

    Implicit Differentiation of Multivariable Functions

    x^2+y^2+r^2-2s=13=0 x^3-y^3-r^3+3s+59=0 How do I find the partial derivatives of x(r,s) or y(r,s) implicitly? I tried implicit differentiation and I got 2 different answers for either. Can someone show me any of the 4 derivatives step-by-step?
  46. C

    Implicit and inverse function theorem

    Lately, we've been going over these two theorems in class. I have a few questions to put forth. 1) I know that in lower spaces, an inverse of a function exists locally (say around a point G) if it does not attain it's max/min at G (i.e. if f'(G) doesn't equal 0). Now, with the inverse...
  47. G

    Another implicit differentiation Problem

    The question I'm having trouble with is as follows: Given that siny = 2sinx show that: a) (dy/dx)^2 = 1+3sec^2(y), by differentiating this equation with respect to x show that b)d^2y/dx^2 = 3sec^2ytany and hence that c) coty(d^2y/dx^2) - (dy/dx)^2 + 1 = 0 Part (c) is straight forward and...
  48. D

    Finding the Second Derivative Using Implicit Differentiation

    Can someone check my answer (I am trying to find the second derivative) for any mistakes? I have looked it over many times, and I've realized that my second derivative is not correct, but I cannot figure out why. Thank you. \sqrt{x} + \sqrt{y} = 1 \frac{1}{2\sqrt{x}} +...
  49. M

    Implicit differentiation problem

    hello everyone I'm stuck! anyone have any ideas? I'm suppose to find dz/dx and dz/dy with implicit differentation. This is calc III! http://img221.imageshack.us/img221/4000/lastscan4ou.jpg
  50. M

    How to Use Implicit Differentiation to Solve for y' in xy^1/2 = 1 + x^2y

    How do I correctly differentiate this using implicit differentiation: xy^1/2 = 1 + x^2y I got to here before I started wondering how I would properly isolate y': ((1/2xy)^-1/2)y + y'x = 2xy + x^2y' How far off the beaten path...
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