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. M

    Implicit Differentiation: Finding dy/dx for cos(y^2) = x^4

    Find \frac{dy}{dx} given cos(y^2) = x^4 Is this correct: 1. cos(y^2) = x^4 2. -sin(y^2) \times 2y \frac{dy}{dx} = 4x^3 3. \frac{dy}{dx} = \frac{4x^3}{-2sin(y^2)}
  2. C

    Derivatives of implicit functions

    If you are given the derivative of an implicit function as y' = \frac{y}{2y+x} how would you find all points (x,y) such that the slope at those points is 1/2? Ok so I did: \frac{y}{2y+x} = \frac{1}{2} and got x = 0. So if I substitute x = 0 back into the original equation I get (0...
  3. C

    Implicit Differentiation Problem - Check my work?

    Implicit Differentiation Problem -- Check my work? I've worked it -- can someone just check my work? Problem: xcosy+ycos=1 My work: [x (d/x)cosy + cosy (d/dx)x] + [y (d/dx)cosx + cosx (d/dx)y] = (d/dx) 1 -xsiny (dy/dx) + cos y - ysinx + cos x (dy/dx) = 0 -xsiny (dy/dx) + cos y...
  4. A

    Implicit Differentiation issue

    Doing fine until I reached a trig function where I know I've done the work correctly but the answer does not match up exactly with the one in the back of the book. \sin(x^2y^2)=x I do the work using product and chain rule \cos(x^2y^2)(2xy^2+2x^2yy')=1 2xy^2+2x^2yy' = \frac {1}...
  5. C

    Implicit Differentiatoin question

    Hello all. I am given the equation (x^3 y^4)^5 = x-y my derivative is 1/ (15x^2 20y^3(x^3y^4)^4)=y' But the book says: y'=(1-15x^14y^20)/(1+20x^15y^19) Where did I go wrong? Thanks so much Chris
  6. I

    Implicit Differentiation

    f(x,y,z,u,v)=xe^y+uz-\cos v=2 g(x,y,z,u,v)=u\cos y+x^2v-yz^2=1 I need to find u_z. When I try to do it by implicitly differentiating and solving the equation, I get 2 contradictory answers. If I try the formula, i.e. f_z + f_uu_z + f_vv_z = 0 g_z + g_uu_z + g_vv_z = 0 I get an answer...
  7. I

    Deriving the Implicit Equation of a Rotated Spheroid

    A spheroid is defined by: x2/a2 + y2/b2 + z2/a2 - 1 = 0 (equation 1) where a and b are the semi-major and semi-minor axes, respectively. If you have any two of x,y,z-values, you can solve for the third, simply by rearranging the above equation: x = +/- sqrt(1 - y2/b2 -...
  8. R

    Second Partial Derivatives of Implicit Functions

    I have been reviewing Calculus and have tripped up on figuring out to calculate the 2nd partial derivatives of imlicit functions. Kaplan and Spiegel give a cursory treatment to the subject in both of their "Advanced Calculus" books. Simply repeating the methods used to calculate the 1st...
  9. J

    Implicit Partial Differentiation

    If there is such a thing. I need to find \partial z / \partial x given x + y + z = \cosh xyz. I've never seen the likes of this before and I haven't a clue where to start. Would a reasonable start be to take \partial /\partial x of both sides? If so, it seems like I'm going to end up with an...
  10. mattmns

    Partial Implicit Differentiation

    Just wondering if I did this right: Here is the question: find \frac{\partial z}{\partial x} of \frac{x^2}{9} - \frac{y^2}{4} + \frac{z^2}{2} = 1 Now I put the \frac{\partial z}{\partial x} on both sides then got. \frac{2x}{9} - 0 + z \frac{\partial z}{\partial x} = 0 So...
  11. C

    Finding the Derivative of Implicit Functions

    Use Implicit Differenciation to find y' f tan(x^2 + y^2) = sec(xy)
  12. Orion1

    Implicit Differentiation: What Am I Doing Wrong?

    Implicit Differentiation...Help! Find \frac{dy}{dx} by Implicit Differentiation: \tan(x - y) = \frac{y}{1 + x^2} \frac{d}{dx} (\tan(x - y)) = \frac{d}{dx} \left( \frac{y}{1 + x^2} \right) \sec^2 (x - y) \cdot \left( 1 - \frac{dy}{dx} \right) = \frac{(1 + x^2) \frac{dy}{dx} - y...
  13. Jameson

    Partial Derivatives Instead of Implicit

    On MathWorld's site, they said that (\frac{\partial{y}}{\partial{x}}){_f} = -\frac{(\frac{\partial{f}}{\partial{x}})_{y}}{(\frac{\partial{f}}{\partial{y}})_{x}} So can this method be used instead of implicit differentiation? Will I get the same result? This seems kind of like a...
  14. C

    Theorem of implicit functions help

    hi guys, im a little stuck at the moment trying to answer the follwing calculus question, can anyone help me please. if x=a(theta-sintheta), y=a(theta+sintheta), find dy/dx and d^2y/dx^2 at the point where theta=pi/2. and given that dy.dx=(dx/dy)^-1, find a fomula for d^2x/dy^2 in terms...
  15. G

    Find y'' by implicit differentiation

    Find y'' by implicit differentiation x^4+y^4=1 i found that y'=-x^3/y^3 y''=[(y^3)(-3x^2)-(-x^3)(3y^2)(y')]/(y^3)^2 y''=[-3x^2(y^3)-(-x^3)(3y^2)(-x^3/y^3)]/y^6 then i am stuck...please help...thanks
  16. K

    How to check implicit differentiation results with calculator?

    What would be the best way to check the result of implicit differentiation with a TI calculator?
  17. K

    Checking implicit differentiation answers?

    How would you check your answer using a Ti86 for implicit diferentiation problems? I was looking through some source code at ticalc.org and found this tidbit for an implict differentiation section: (after given a point x and y, with function F1) If der1(F1,y)==0 [exit] else...
  18. K

    Can Implicit Differentiation Be Used to Solve for dy/dx?

    Hi, here is my problem. I think it has something to do with me not completely understanding implicit differentiation. I have to find \frac{dy}{dx} of x^2+5yx+y^5=8 To do this, I differentiated the x^2 as 2x then I used the product rule to differentiate 5xy into 5y + \frac{dy}{dx} * 5x. I...
  19. E

    Implicit differentiation and a bicorn curve

    Hey all. There's a question I seem to be stuck on involving implict differentiation. Here it is: The curve called a bicorn has the equation (x^2+8y-16)^2=y^2(16-x^2) Verify by implicit differentiation that its tangent lines at the points (0,4) and (0,\frac{4}{3}) are horizontal. Do by...
  20. T

    Implicit diff (2 var), error or what?

    i am working on a homework assignment. it's easy, or, so i think... Given. 3x^2 - xy^3 + sin(x^3 - y) = 4 Find \frac{dy}{dx} not a problem. i ended up with \frac{dy}{dx} = \frac {6x - y^3 + 3x^2 cos(x^3 - y)}{3xy^2 + cos(x^3 - y)} using implicit differention. now...
  21. C

    Verification of Implicit Differentiation

    Help with Implicit Differentiation Hello all If we are given \cos xy = 2x^2 - 3y find \frac {dy}{dx} So the derivative of \cos xy is - sin(xy)(x)(\frac{dy}{dx} + y) The derivative of the RHS is 4x - 3 \frac {dy}{dx} Hence \-sin(xy)(x)\frac{dy}{dx} + y = 4x - 3 \frac...
  22. D

    How do I find v' for functions v=x+y using implicit differentiation?

    Say we have two functions of x, v and y, such that v=x+y. How can I find v'?
  23. C

    How do I find the derivative of y using implicit differentiation?

    Hello all Given: x^2 + xy + y^2 - 7 = 0, solve for y using the quadratic forumula. Then find dy/dx at P(1,2) from a function of the form f(x). My solution: y = -x (+/-) sqrt( x^2 - 28) / 2. I am not sure if this is correct. After solving for y, do you have to implicitly take the...
  24. R

    Finding y' by implicit differentiation SO HARD FOR ME help~

    Find dy/dx by implicit differentiation. (Meaning find the derivative of y.. so y' = ??) cos(x-y)=xe^x Please help and show step by step.. the final answer should be.. y' = 1 + [(e^x)(1+x)]/[sin(x+y)] Thanks a bunch. :smile:
  25. T

    How Do You Derive the Slope from the Equation 2xy^2+xy=y?

    taking derivitive of 2xy^2+xy=y split up to using product rule 2x----2 y^2---2y\frac{dy}{dx} 4yx\frac{dy}{dx}+2y^2+y+x\frac{dy}{dx}=\frac{dy}{dx} \frac{dy}{dx} (4x+x-1)=-2y^2-y \frac{dy}{dx}= \frac{-2y^2-y}{4xy+x-1} i am trying to figure out the slope of the equation...
  26. Z

    Use implicit differentiation to find the points of tangency

    Please Help! Ok, I am having problem with an Implicit differentiation problem... Two tangent lines to the hyperbola 9x^2 - y^2 =36 intersect at the y-axis. Use implicit differentiation to find the points of tangency. Ok so i implicitly differentiated this function and i came up with y'=...
  27. C

    How do you find the second derivative using implicit differentiation?

    Hi all: How would you find the second derivative of 2x^3 - 3y^2 = 8? I know the first derivative is x^2 / y. Would I use the quotient rule, or would I use some type of substitution and then use the product rule? Any help is greatly appreciated! Thanks
  28. B

    Having a hard time with 2 implicit equations

    First off, I must say I truly enjoy these forums, though I tend to shy away from Calculus. Calculus and I just don't seem to get along, which is strange since I really enjoy other forms of math. On to the questions. Note, these 3 are the ones I just can not seem to work through on an...
  29. A

    Inverse and Implicit Function theorems

    Hello all, Can somebody give a 'simple to understand' proof for the 'Implicit Function Theorem' and the 'Inverse Function Theorem',their significance, applications and some examples. I have tried some books on Multivariable calculus, but in vain... :cry:
  30. D

    Finding Turning Points on a Curve with Implicit Differentiation

    find the turning points on the curve with equation y^3 + 3xy^2 - x^3 = 3 I'm confused bout the 3xy^2 mostly, i kno that xy goes to (y+x dy/dx) but not sure what 2 do bout the power...
  31. A

    Implicit Differentiation: Finding Points on an Ellipse Tangent to a Box

    My TA did not get the chance to go over this problem. I know that I'm supposed to differentiate both sides of the equation. But I have not the slightest idea what to do after that. I was told that I am supposed to get out 4 points that lie on the ellipse and the sides of the box are tangent...
  32. C

    Implicit Differentiation Problem (stuck at a point)

    Here is the problem: Find the equation of the line which is tangent to the curve at the point (1,3): 8x^3y^2 + x^2y^5 + 6 = 4y^4 - 3x^4 Here is what I've done so far (I'm stuck now): (24x^2)(y^2) + (8x^3)(2y dy/dx) + (2x)(y^5) + (x^2)(5y^4 dy/dx) = (16y^3 dy/dx) + 12x^3 Where do...
  33. M

    P implicit vs. explicit function of time

    As part of proving that d/dt <p> = <-Del V> you have to use the fact that < dp/dt> = 0 when p is not an explicit function of time. I'm not clear on what this means. Any insights to share? From classical mechanics, if there is a potential V = -k/r then there will be a force on a particle...
  34. A

    Can someone explain Implicit Differentiation and Related Rates in Calculus?

    Hi. I'm taking a Calculus course right now and I simply cannot understand Implicit Differentiation or the Related Rate problems. My textbook does not do a good job explaining it. It is a very accelerated class and I cannot get it and I need to know it in two days for a mid term. I just don't...
  35. G

    Problem that uses implicit differentiation

    Can u please give me an example of a problem that use implicit differentiation to find the derivative of an inverse function and of relations. Plz also explain to me what relation means in this case. Thanks also please give me an example that use the definite integral to compute accumulated...
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