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. D

    Equation of tangent - Implicit or Partial DifferentiatioN?

    Homework Statement Need to find the tangent to the curve at: e^(xy) + x^2*y - (y-x)^2 + 3 I just implicitly differentiate the expression to find the gradient and then use the points given to find the equation, right? Or does this involve partial differentiation? Homework Equations...
  2. B

    What is the process for implicit differentiation?

    I am reading about this topic, and I came across this sentence "Remember, every time we want to differentiate a function of y with respect to x, we differentiate with respect to y and then multiply by dy/dx." What exactly does this mean?
  3. A

    Show that this implicit equation can be expressed as the vector equation

    Homework Statement Show that x^3+y^3=9xy can be expressed as r(t) = \frac{t}{1+t^3} \hat{i} + \frac{t^2}{1+t^3} \hat{j} The Attempt at a Solution I only know how to convert parametric/vector equations to Cartesian, not the other way around which is what I'm pretty sure I have to...
  4. I

    I seem to lack an implicit differentiation technique, should be a quick fix

    Homework Statement u = x^u + u^y Find the partial derivatives of ##u## w.r.t. ##x, y##.Homework Equations Only the one. The Attempt at a Solution I've attempted reducing the problem using logs, but the resulting equations seem no more tenable to me. I'm sure there is a nice trick... it...
  5. R

    Implicit Methods for Drag-Dependent Acceleration in Euler Integration

    I'm trying to write a code to implement he backwards Euler method to integrate the equation of motion. The sticking point seems to be that the acceleration is due to drag, and thus is dependent on the new position and velocity. I understand the method to be: v_{i+1}=v_{i}+a_{i+1}δ...
  6. E

    Implicit Differentiation Practice

    Homework Statement find dy/dx if x=1/1 + t and y = t2/1 + t Homework Equations dy/dx = (dy/dt) / (dx/dt) The Attempt at a Solution dx/dt = 1 * (1 + t )^-1 dx/dt = -1/ (1 +t)^2 dy/dt = t^2 * (1 +t)^-1 dy/dt = -2t/(1+ t)^2 therfore dy/dx = -2t/(1 + t)^2 / -1/(1 +...
  7. MathematicalPhysicist

    Mathematica Letting mathematica compute taylor expansion of implicit function.

    I have the next function: z^3-2xz+y=0 and I want to find taylor expansion of z(x,y) at the point (1,1,1), obviously I need to define F(x,y,z) as above and use the implicit function theorem to calculate the derivatives of z(x,y), but I want mathematica to compute this to me. I tried the Series...
  8. C

    Implicit differentiation problem.

    Homework Statement Find dy/dx in terms of x and y if.. x2-√(xy)+y2=6 Homework Equations The Attempt at a Solution so I started by.. x2-√(xy)+y2=6 deriving the LHS 2x+2y(dy/dx)-1/2(xy)-1/2(1(y)+x(dy/dx)) Simplifying the last term...
  9. S

    Implicit differentiation question?

    Homework Statement use implicit differentiation to find an equation of the tangent line to the curve a the given point. y^2(y^2-4) = x^2(x^2-5) at (0,-2) Homework Equations y^2(y^2-4) = x^2(x^2-5) The Attempt at a Solution I got dy/dx to be (3x^2-10x)/(4y^3-8y) but...
  10. B

    Implicit partial differential for unknown function F(x,y,z)=0

    Homework Statement F(x,y,z)=0 x=a(y,z) y=b(x,z) z=c(x,y) what does \frac{\partial c}{\partial x} \frac{\partial b}{\partial z} \frac{\partial a}{\partial y} equal. Homework Equations maybe you could tell me? The Attempt at a Solution i've spent hours and hours and pages...
  11. M

    Implicit Function Theorem

    Let F(x, y) = f(y − x + g(y + x)), where f(u) and g(u) are sufficiently differentiable functions of a single real variable. If, in a neighbourhood of (x, y) = (a, b), the equation F(x, y) = 0 defines a function y(x), state the condition(s) on f and g so that y′(x) exists in a neighbourhood of x =...
  12. F

    Implicit differentiation - major confusion

    Hello. I know how to do implicit differentiation taught in calculus 1, but I'm confused by something regarding it. Take the example: y3+y2-5y-x2=4 If we do implicit differentation we get: 3y2(dy/dx)+2y(dy/dx)-5(dy/dx)-2x=0 dy/dx=2x/(3y2+2y-5) Now, it makes sense how to...
  13. S

    Implicit isomorphism involved in extension/sub fields/structures?

    Implicit isomorphism involved in extension/sub fields/structures? This has been bugging me for a while. I'm pretty sure I'm correct but I'd just like to verify to put my mind at ease. I'd like to know if there is an implicit isomorphism involved when we say, for example, F is a substructure of...
  14. S

    Implicit Differentiation concept help?

    Homework Statement Its not homework, i have the answer I am just having a hard time wrapping my head around the concept of differentiating implicitly defined functions. the question was: x^3+y^3=3xy, find the equation of the tangent line at the point (3/2,3/2). Homework Equations...
  15. B

    Implicit Differentiation Question

    Homework Statement 2x^{3}-3x^{2}y+2xy^{2}-y^{3}=2 Homework Equations The Attempt at a Solution 6x^{2}-(6xy+3x^{2}y')+(2y^{2}+4xyy')-3y^{2}y'=0 y'=\frac{-6x^{2}+6xy-2y^{2}}{-3x^{2}y+4xy-3y^{2}} My text's solution is the same answer but with every every term having the...
  16. I

    Find dV for V= 200/(x2 + y2 )1/2 when x=2 and y=1 - Homework Help

    Homework Statement The electrical potential can be described by the following equation: V= 200/(x2 + y2 )1/2 find dV when x=2 and y=1 Homework Equations n/a The Attempt at a Solution dv/d(x,y) = ∂/∂x + ∂/∂y =200/(x2 + y2)(1/2) +200/(x2 + y2)(1/2) replace variables with C where...
  17. B

    Implicit Derivation: Finding 2nd Derivative of y^2 = x^3

    Homework Statement I am suppose to find the second derivative implicitly of the function y^2 = x^3. I find the first derivative to be dy/dx = 3x^2/2y, but shortly find myself having difficulty in the second derivation. My steps for the second derivative is in the file attached; there are a few...
  18. L

    Implicit vs non implicit diff. (line passing through point & tangent to function)

    Homework Statement Find the equations of the lines that pass through (0,0) and are tangent to x^2 - 4x + y^2 + 1 = 0 My confusion I've been given a problem of this sort recently, except now it involves implicit differentiation. I know "how" to get to the correct answer. I just...
  19. D

    Why is there a dx/dx in the Implicit Differentiation rule?

    I tried deriving this one on my own and I'm just not understanding where the dx/dx term comes from. I'm looking dy/dx. Starting with F(x,y) = 0: \frac{\partial{F}}{\partial{x}}\frac{dx}{dx} + \frac{\partial{F}}{\partial{y}}\frac{dy}{dx} = 0 It seems redundant to say dx/dx when it turns out to...
  20. L

    3(x^2 + y^2)^2 = 25(x^2 - y^2) implicit differentiation

    Use implicit differentiation to find the slope of the tangent line to the curve at the specified point. 3(x^2 + y^2)^2 = 25(x^2 - y^2) ; (2,1) This is where I'm stuck: I know how to get up to the first equation...and I know how to get to the final answer from the second equation, but I...
  21. N

    Implicit differentiaion using the number e (lon-capa)

    Implicit differentiaion using the number "e" (lon-capa) Hello~ :3 this is my first time posting here, so I hope I didn't do anything wrong. I'm currently in Calculus 1, university level, and I have to enter all my answers using lon-capa (evil evil program). In lon-capa: *= multiplication...
  22. T

    MATLAB Matlab program using implicit Finite Difference

    Hello, I need help writing a MATLAB program to solve a heat transfer problem implicitly. For some reason this is very confusing to me. The problem is stated below. Any help is greatly appriciated. Let me know if you need a little more info. I need to write a program to solve this...
  23. E

    Comp Sci Fortran 95 - ADE Implicit Scheme Tridag. Matrix

    Homework Statement The problem statement is to use Fortran 95 to code a forward time, centered space numerical solution to the 1-D (x-direction) Advection-Dispersion Equation: dc/dt = u(dc/dx) + D(d2c/dx2) - kc where c is the concentration of contaminant, u is the advecting velocity, D is...
  24. A

    Finding Stationary Points on Implicitly Differentiated Curves

    Homework Statement Find the coordinates of the stationary points on the curve: x^3 + (3x^2)(y) -2y^3=16 Homework Equations Stationary points occur when the first derivative of y with respect to x is equal to zero The Attempt at a Solution I implicitly differentiated the...
  25. D

    Is the Implicit Function Theorem Applicable to F=K?

    Can anyone check if this argument works. I just made up this problem to check if I understand what's going on. Consider F = xy*e^x + y*e^y = K. I want to see if there is a unique solution. Fy = x*e^x + y*e^y + e^y. Since we are on the surface x*e^x + y*e^y = K. So if K >= 0 then...
  26. T

    Chain rule for implicit differentiation

    I have derivative dx/dt = y(u(t)) * z(u(t)) + u(t) Now, what is dx/du ? I know the chain rule should help, but I am stuck :-(
  27. J

    Implicit exponential differentiation?

    Homework Statement Find an equation of the tangent line to the curve xe^y+ye^x=1 at point (0,1). Homework Equations I do not recall seeing the Implicit Function Theroem before, I even went back in my book (Stewart Calculus 6th) to check. I found this post but it does not help me...
  28. nomadreid

    Are implicit and partial differentiations related?

    I was working two different but superficially related problems, and noticed that if I did something that is generally not allowed, the results were connected by a negative sign. My questions are whether this will always turn out this way, and if so, why. The two problems were (A)...
  29. 5

    Implicit partial differentiation functional det[]

    Hello! I was wondering how I could find the following derivatives from the given function using Jacobian determinants. f(u,v) = 0 u = lx + my + nz v = x^{2} + y^{2} + z^{2} \frac{∂z}{∂x} = ? (I believe y is constant, but the problem does not specify) \frac{∂z}{∂y} = ? (I...
  30. M

    Implicit second order partial derivative

    Homework Statement Given that the surface (x**5)(y**2)+(y**5)(z**3)+(z**3)(x**2)+4xyz=7 has the equation z=f(x,y) in a neighbourhood of the point (1,1,1) with f(x,y) differentiable, find the derivatives (∂**2f)/(∂x**2) at (1,1) Homework Equations The Attempt at a Solution I...
  31. S

    What is the cause of these error- symbol 'x1' has no implicit type?

    Hi all, i am beginner in fortran and linux. I have wrote codes for 1d scalar wave as below: SUBROUTINE fd1d (x_num,x1,x2,t_num,t1,t2,c,u_x1,u_x2,u_t1,ut_t1,u) implicit none integer (kind=2) t_num,x_num,t,x real (kind=4)...
  32. Twinflower

    Derivative of y with Respect to x: Solving Implicit Differentiation Problems

    Homework Statement Derivate y with regards to x, \frac{dy}{dx} 2xy+y^2-4x=10 Homework Equations The Attempt at a Solution I am not very good at differentiation, so I haven't got off to a good start, really. I guess I can differentiate y^2 and -4x "the usual way", but I am kind of stuck with...
  33. T

    Numerical solution of continuity equation, implicit scheme, staggered grid

    Hi! I'm trying to implement an implicit scheme for the continuity equation. The scheme is the following: http://img28.imageshack.us/img28/3196/screenshot20111130at003.png With \rho being the density, \alpha is a weighing constant. d is a parameter that relates the grid spacing to the...
  34. nomadreid

    Implicit differentiation gives too many stationary points

    If, with y a function of x, I have the equation x2-5xy+3y2 = 7, then by implicit differentiation, I get that dy/dx = (2x-5y)/(5x-6y). This equals zero everywhere on the straight line y=(2/5)x except at the origin. This would seem to indicate stationary points everywhere on that line, which is...
  35. P

    Implicit Differentiation (for a normal derivative) with Multivariable Functions

    Homework Statement z^{3}x+z-2y-1=0 xz+y-x^{2}+5=0 Define z as a function of x, find z'. Homework Equations I guess the two equations above... The Attempt at a Solution Well, I just differentiated the first one with respect to x and got: 3z^{2}xz'+z^{3}3+1=0 z' = \frac{-1-z^{3}{3z^{2}x}...
  36. J

    Implicit Differentiation: How Can I Solve for y' in e^(x/y)=x-y?

    Homework Statement Find y' in e^(x/y)=x-y 2. The attempt at a solution I tried to differentiate it by changing it so that there would be a natural log (as seen in my attachment). However the end result is not the same as the answer key. How the answer key did it was they used the...
  37. T

    Implicit Differentiation and Orthogonal Trajectories

    Homework Statement Q 50: The ellipse 3x2 +2y2 = 5 and y3 = x2 HINT: The curves intersect at (1,1) and (-1,1) Two families of curves are said to be orthogonal trajectories (of each other) if each member of one family is orthogonal to each member of the other family. Show that the families of...
  38. F

    Solving Implicit Equations for y in Terms of x

    How do you go about solving implicit equations for y in terms of x that look like these? y2 + yx = 1 and y3 + yx = 1 or even more complicated implicit equations. I'm taking AP Calculus AB this year and am just curious how this is done. Here are the solutions from Wolframalpha...
  39. K

    Solving implicit equation of a plane

    Hello, Can someone tell me if what I am doing is correct. I am trying to solve this equation of 0 = n ( p - p0 ) where n, p, and p0 are points on the plane. Assume this is a 2D dimension (x,y). Is this the correct way of calculating this equation. 0 = nx (px - p0x) + ny (py + p0y)...
  40. J

    Where am I going wrong on implicit differentation?

    Homework Statement Determine the equation of a tangent line to a curve at the given point. 3x2+xy+2y2=36 , P(2,3) Homework Equations The Attempt at a Solution 3x2+xy+2y2 = 36 finding the derivatives of each term I get: 6x+xy'+y+4yy' = 0 xy'+4yy' = -6x-y y'(x+4y) =...
  41. B

    Implicit (Backwards) Euler's Method

    How would one go about solving for one variable for an implicit Euler's method such as this: I am completely lost...all I know is the value of U and dT Un+1 = Un + (dU/dT)|n+1dT Vn+1 = Vn + (dV/dT)|n+1dT
  42. C

    Second derivative using implicit differentiation

    Homework Statement x^6 + y^6 = -6 I have to prove that y'' = 30x^4/y^11 Homework Equations The Attempt at a Solution Using implicit differentiation: 6x^5 + 6y^5 dy/dx = 0 6y^5 dy/dx = -6x^5 dy/dx = -x^5/y^5 Quotient Rule: [(y^5)(-5x^4) - (-x^5)(5y^4 dy/dx)] /...
  43. C

    Implicit and explicit solution for a given initial-value problem

    The equation is: x2 dy/dx= y - xy IC (initial conditions): y(-1) = -1 (This is used to solve for C) Must first separate the variables x and y and then integrate them and solve for y, but I got stuck... x2dy = (y - xy)dx x2dy = y(1-x)dx dy= y(1-x)dx/x2 <-- not sure what to do with the y now...
  44. J

    Finding an implicit, general solution to a homogeonous differential equation

    Homework Statement Find an implicit, general solution to: dy/dx = (6x - 4y) / (x - y) with x > 0. Homework Equations The Attempt at a Solution dy/dx = (6x - 4y) / (x - y) x(dv/dx) + v = (6 - 4v) / (1 - v) [(1-v) / (v-3)(v-2)] dv = dx / x \int dv/(v-2) - 2\int...
  45. A

    Implicit differentiation, find y''

    the problem is to find y'' or d2y / d2x the equation is y2 = x2 first i found the first derivative dy/dx = 2x / 2y = x / y then i found the second using the quotient rule and got y'' = (y - x(dy/dx)) / y2 i plugged in y' into y'' and got y'' = (y - (x2/y)) / y2 but then I am...
  46. L

    Differential Equations Verifying Implicit Solution

    I'm given: 1. \frac{dX}{dt}=(X-1)(1-2X) 2. ln(\frac{2X-1}{X-1})=t and asked to verify that it is an implicit solution to the first order DE given. I successfully derived the second equation there to get: \frac{dX}{dt}=\frac{-1}{(2X-1)(X-1)} So now what? I tried several things and...
  47. T

    Differential Equations, implicit equation solving

    Homework Statement (b) Find the differential equation for which –4xy3 + 4xy3sin(x) = –1 is an implicit solution on the interval (0, pi/2). Write your answer in the form dy/dx = f (x,y) where f (x, y) depends on both x and y. The Attempt at a Solution I'm not too sure on how to go about...
  48. M

    Interesting plot of an implicit equation with square root of signum functions

    Hello, recently, I have found so called Batman graph and equation, which basically plots Batman sign. I have even found Python code that plots this implicit equation. Code can be found here: . And the equation is given here: [url]http://i.imgur.com/CNy9J.jpg If you look closelx...
  49. A

    Implicit Differentiation: Finding d^2y/dx^2 at a Given Point

    Homework Statement find the d^2y/dx^2 if y^3 + y = 2 cos x at the point (0,1) Homework Equations The Attempt at a Solution my dy/dx = (-2 sin x)/(3y^2 + 1) I don't know how to find d^2y/dx^2? And when and how will I use the oint (0,1)?
  50. L

    Derivative implicit differentiation problem

    Homework Statement Consider the curve satisfying the equation 2(x+1)^(tanx)=(y^2)cosx+y and find dy/dxHomework Equations (tanx)'=sec^2x (lnx)'=1/xThe Attempt at a Solution I've tried taking the natural log of both sides and then taking d/dx of both sides but something seems to go wrong with...
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