Implicit differentiation Definition and 279 Threads

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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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 :-(
  7. 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...
  8. 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}...
  9. 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...
  10. 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...
  11. 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)] /...
  12. 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...
  13. 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)?
  14. 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...
  15. Femme_physics

    What is the use of implicit differentiation in solving complex equations?

    http://img838.imageshack.us/img838/8007/given01.jpg http://img220.imageshack.us/img220/52/given2r.jpg Actually that should be the square root of -2X
  16. B

    Related rates and implicit differentiation

    Homework Statement A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30 degrees. At what rate is the distance from the plane to the radar station increasing a minute later? Homework Equations a2+b2=c2The...
  17. N

    Trouble With Understanding Implicit Differentiation

    I don't think I fully understand implicit differentiation. I have read my textbook and watched many videos, and I think I will get an A on my test on this solely by memorizing the rules, but I would really like to understand this topic. From what I know, you are supposed to use implicit...
  18. 1

    How to Treat Y When Differentiating Implicitly?

    Homework Statement x^{3}y + y^{3}x = 30 Homework Equations - The Attempt at a Solution My understanding is that I'm supposed to take the derivative of each side and try to solve for y'. But, I don't know how to treat the y when I am differentiating it, to a power or whatever...
  19. QuarkCharmer

    Implicit Differentiation, and the Chain Rule

    Homework Statement Use implicit differentiation to find dy/dx 2x^3+x^2y-xy^3 = 2 Homework Equations Chain Rule et al. The Attempt at a Solution My questions is this. When deriving something like xy^3, apply the product rule to get 1y^3 + x\frac{d}{dx}y^3 I am confused on...
  20. F

    Implicit Differentiation with Mathematica

    Homework Statement I tried using D[x = y - y^2, x, NonConstants -> {y}] and it keeps telling me that y - y^2 is not a valid variable.
  21. S

    Questions involving implicit differentiation,

    Homework Statement Q1. using implicit diff to find dy/dx when x^2 y + 6xy^2 = 5x-2 Q2. Find max and min values of y= x^3 -3x^2 -6x + 7 on the interval -3<=x<=5 Q3.Find the exact values of the x coordinate of the points of inflexion on the graph of y = 2x^4 +3x^2 +x +5 Q4. a red car is...
  22. P

    Calculus problem involving implicit differentiation.

    Good afternoon, This is not actually a homework question; it's for self-study. I'm reading a Calculus book, and one of its exercises asks the following: If xnym = (x+y)n+m, show that xDxy = y (where Dxy is the derivative of y with respect to x). The only way I could think of to get the correct...
  23. R

    How Does Implicit Differentiation Apply to Polynomial Equations?

    Question: y6 + 6 (x^2+4)6 = 9 6y5 .dy/dx . 6(x2 + 4)5 . (2x) = 0 6y5 .dy/dx = -6(x2+4)5 .(2x) dy/dx = 6y5 / -6(x2 + 4)5 .(2x) dy/dx = 6y5 / 12x(x2 +4)5 Although the answer is ment to have y5 as the numerator, not 6y5? ------------- Another Q. [Simplifying result from...
  24. H

    Problem involving implicit differentiation over an ellipse

    Homework Statement So here's a question from my textbook 'Calculus: Concepts and Contexts' 2nd ed. by James Stewart. This is section 3.6 # 54 We have Cartesian coordinates set up with an ellipse at x^2 + 4y^2 = 5 To the right of the ellipse a lamppost (in 2D!) stands at x=3 with...
  25. C

    What is the Implicit Derivative at a Given Point?

    Homework Statement "Find dy/dx at the given point by using implicit differentiation" x2y + y2x = -2 at (2, -1) and (x+y)3 = x3 + y3 Homework Equations The Attempt at a Solution 1) x2(dy/dx) + y(2x) + y2(1) + 2y(dy/dx)(x) = -2 x2(dy/dx) + 2xy + y2 + 2xy(dy/dx) = -2...
  26. Kawakaze

    Beginner implicit differentiation

    Please go easy on me, 2 days ago I didnt even know what implicit differentiation was. Homework Statement If x tan y − y tan x = 1, use implicit differentiation to determine dy/dx, expressing your answer in the form dy/dx = f(x, y), The Attempt at a Solution Differentiate first...
  27. T

    The Asteroid (implicit differentiation)

    Homework Statement Calculate the length of the graph/equation: x^(2/3) + y^(2/3) = a^(2/3) The graph is formed as an s.c. asteroid, almost like a diamond. It seems to be some sort of modified unit circle. Homework Equations The length of the graph between x1 and x2 can be described as L=∫...
  28. J

    How Is Implicit Differentiation Applied to Solve 5y^2 = 4x - 3/4x + 3?

    Homework Statement Given 5y^2 = 4x - 3/4x + 3 Homework Equations is it permissable to say this is equal to y^2 = 4x -3 /5(4x + 3) and then 2y(dy)/(dx) = what the right side equals thru using the quotient rule? I know the answer is dy/dx = 12/5y(4x + 3)^2 but I don't know how to...
  29. O

    MATLAB Implicit differentiation in matlab

    I have a implicit differential equation Re(u*ux-u^2)=0 ( It was larger equation but i simplified it here). I wrote down my function in m.file as follows: function Z=fun_imp(x,u,ux); Re=25; Z=0; Z=Re*(u*ux-2*u); Before going to solving differential equation to find out consistent...
  30. H

    Implicit Differentiation - 2nd derivative

    Homework Statement Find d2y/dx2 in terms of x and y for the following equation: xy + y^2 = 1 COMPLETELY SIMPLIFY Homework Equations dy/dx The Attempt at a Solution so i get -y/(x+2y) for the dy/dx. When I try to find the 2nd derivative and plug in dy/dx, I get...
  31. Telemachus

    Second derivatives using implicit differentiation

    Hi there. Well, I wanted to know how to find the second derivatives of a function using implicit differentiation. Is it possible? I think it is. I think I must use the chain rule somehow, but I don't know how... I'm in multivariable calculus since the function I'm going to use could be seen as a...
  32. A

    Implicit differentiation help again

    implicit differentiation help :) again! Homework Statement Use implicit differentiation 1) x/(y-x2)=1 and 2) (y2-1)3=x2-y The Attempt at a Solution 1) x/(y-x2)=1 => [(y-x2)(1)-(x)((1*dy/dx)-2x)]/[(y-x)2]2=0 => y-x2-x(dy/dx)+2x2=[(y-x)2]2 I think I'm going to stop here because I'm pretty...
  33. A

    Horizontal tangents via implicit differentiation

    Homework Statement Find the points (if any) of of horizontal tangent lines on : x2 + xy + y2 = 6 Homework Equations n/a The Attempt at a Solution So far I've concluded that I must find the points at which dy/dx = 0. I've solved for dy/dx and arrived at dy/dx = (-2x-y)/(x+2y)...
  34. C

    Complicated implicit differentiation

    Homework Statement If (sqrt x) + (sqrt y) = 11 and f(9)=64 ---> find f '(9) by implicit differentiation The Attempt at a Solution I keep getting lost in my work here... first, taking derivative of both sides: d/dx ((sqrt x) + (sqrt y)) = d/dx (11)...
  35. D

    Implicit Differentiation and Complex numbers concept question

    I just started learning Implicit Differentiation and came across an issue. I took the derivative of the circle function: y2 + x2 = 1 y' = -x / y This all made sense until I solved the circle function for y, which gives: y = \pm\sqrt{1 - x^2} For any x > 1, it's going to be complex. So, does...
  36. H

    Implicit Differentiation Tangent lines

    Homework Statement Where does the graph of 25x^2 + 16y^2 + 200x - 160y + 400 = 0 have a horizontal tangent line. Homework Equations dy/dx dx/dy or something not sure. The Attempt at a Solution Well I know that a horizontal tangent line would mean the slope is zero...but what...
  37. A

    How Do You Isolate dy/dx in Implicit Differentiation?

    implicit differentiation help :) Homework Statement Find dx/dy by implicit differentiation (x2+ y)2+ x2+ xy2= 100Homework Equations The Attempt at a Solution I'm trying to use the chain rule to solve it... i got The derivative ofb(x2+ y)2+ x2+ xy2= 100, with respect to x, is 2(x2+...
  38. A

    Implicit Differentiation: Solving for dx/dy in an Example Problem

    Let me first say i just learned implicit differentiation (earlier today) and i am also new in general to derivatives. I am finding implicit differentiation difficult but i want to understand it before we go over it in class. Homework Statement This is a example in my book. I have been trying...
  39. P

    Implicit differentiation - Help

    Homework Statement find y' given x3+y2+x+y=2 Homework Equations The Attempt at a Solution dy/dx=3x2+2y(y')+1+y' I know that's wrong because my 89 gives it as y'=(-3x2+1)/(2y+1), I don't know how to get there though and what happened to the =2. Thanks
  40. S

    Superior implicit differentiation, prove answer.

    Homework Statement 2XY = Y^2 prove that y''2 = \frac{y^{2}-2xy}{(y-x)^3} EDIT: Sorry, don't know how to insert a space in Latex. Homework Equations The Attempt at a Solution 2y+2x \frac{dy}{dx} = 2y \frac{dy}{dx} \frac{dy}{dx}(2y-2x) = 2y \frac{dy}{dx}= \frac{y}{y-x}...
  41. J

    Implicit Differentiation Shortcut

    Homework Statement differentiate: yx = y / sqrt(x2 + y2) Homework Equations The Attempt at a Solution I solved this problem by taking the ln of both sides and then solving. It seems from the context of the problem set that this was supposed to be easier than that. Am I...
  42. L

    Implicit differentiation, whats going wrong?

    Implicit differentiation, what's going wrong!? Hey people can someone point out to me please where I'm going wrong with part B of this question, can't get it to look the answer in the book, d2y/dx2 = -3(x^6/y^7)- 3(x^2/y^3). My answer is listed below under part B section, but i can't...
  43. P

    Implicit differentiation - Help

    Homework Statement Find d2y/dx2 for cos y + x2 =12 The first derivative I think is correct but the second I am unsure of, this is Q5 of 10 in an assignment and every other question has been far cleaner and easier making me think that I've missed something. We don't have to use implicit...
  44. B

    Implicit Differentiation; Tangent Line

    Homework Statement Find the equation of a tangent line at the curve at point (-3√3, 1) x^(1/3) + y^(1/3) = 4Homework Equations Point-slope: y-1=m(x-1) The Attempt at a Solution I took the derivative of that equation and resulted in -y^(2/3)/x^(2/3) When I tried plugging in x and y to...
  45. S

    Implicit differentiation and optimization

    Homework Statement A conical tent must contain 40\pi ft^{3}. Compute the height and radius of the tent with minimal total surface area. (Include the floor material.) Homework Equations 1. \frac{\pi r^{2} h}{3} = 40\pi 2. \pi r \sqrt{r^{2} + h^{2}} + \pi r^{2} = S 3. \frac {dr}{dh} =...
  46. C

    Finding dy/dx of y=1/(x+y) using implicit differentiation: Step-by-step guide

    can someone show me step by step how to find dy/dx of y=1/(x+y) using implicit differentiation?
  47. D

    Implicit differentiation and the product rule

    An example of implicit differentation in Stewart, 6th ed, p 883, is given as follows: x^3 + y^3 + z^3 + 6xyz = 1 Differentiating to find dz/dx, 3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0 where the product rule was used to differentiate 6xyz with respect to x. Why isn't the...
  48. L

    Implicit Differentiation: Solving for dy/dx

    1. I'm pretty sure I've got this right.. just needing a check. Cheers! 2. x^2+2xy-y^2+x^2=2 3. 2x+2x(dy/dx)+2y-2y(dy/dx)+2x=0 2x(dy/dx)-2y(dy/dx)=-4x-2y dy/dx(2x-2y)=-4x-2y dy/dx=(-4x-2y)/(2x-2y)
  49. T

    Horizontal tangent points (implicit differentiation)

    Homework Statement Find the points on the lemniscate: 2( x^2 + y^2 )^2 = 25( x^2 - y^2 ) where the tangent is horizontal Homework Equations Horizontal tangent: y' = 0 The Attempt at a Solution I got the correct gradient of y' = [ 50x - 8x^3 - 8y^2 ] / [ (8x^2)y + 50y + 8y^3 ]...
  50. T

    What is the Value of dy/dx When y=1 in Implicit Differentiation?

    Homework Statement If y= \ln \sqrt{xy} , find the value of dy/dx when y=1 Homework Equations The Attempt at a Solution \frac{dy}{dx} = \frac{1}{\sqrt{xy}} \cdot \frac{1}{2\sqrt{xy}}\cdot (x\frac{dy}{dx}+y) \frac{dy}{dx}=\frac{1}{x} , when y=1
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