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. Math Amateur

    MHB Analysis of Implicit Differentiation

    I am reading several books on multivariable analysis/calculus and am trying to get a precise and rigorous theoretical understanding of implicit differentiation, including the Implicit Function Theorem ... in particular I am reading: Vector Calculus (Second Edition) by Susan Colley and...
  2. S

    How can I solve this problem using implicit differentiation?

    I unfortunately keep on getting the wrong answer to this problem. I am supposed to find: dw/dy(1/(w^2+x^2)+1/(w^2+y^2)) I attached a picture of how I tried to solve it. Help would be much appreciated.
  3. M

    Differentiation of a sphere -- raindrop evaporating as it falls

    < Moderator Note -- Thread moved from the technical PF Calculus forum > I can't seem to grasp the idea of this problem, any help is much needed. The problem reads, "As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its...
  4. F

    MHB Implicit differentiation difficult

    Given that $y'=\frac{tan(y)}{1-xsec^2(y)}$, find y'' in terms of $x$ and $y$ only. I've done this and checked my work several times but my answer does not agree with wolfram alpha. Sorry for not posting my work, I am a bit busy at the moment. Can someone show the first couple of lines of work...
  5. N

    MHB Learn Implicit Differentiation: Solving for r^2 in $y^2 + x^2 = 0

    Can someone please explain how the result is obtained from the first line $r^2 = y^2 + x^2$ (refer to attached image)
  6. D

    Implicit differentiation; reproducing textbook derivation

    Homework Statement (The fourth equation is the central one) first, we have \frac{1}{r}=\frac{a}{b^2}(1+ecosθ) and b^2=a^2(1-e^2) now using these two, we transform acosψ=ae+rcosθ into (1-ecosψ)(1+ecosθ)=\frac{b^2}{a^2} we want to find dθ/dψ, and the author performs an inplicit...
  7. F

    MHB Question about implicit differentiation

    Say you want to find the slop of a tangent line of the circle x^2+y^2=25 I was following the directions here. I don't completely understand how the derivative of y^2 becomes 2y\frac{dy}{dx}. Shouldn't it become 0 if we are taking the derivative with respect to x? The website explains but to me...
  8. Greg Bernhardt

    What is implicit differentiation

    Definition/Summary The definition of a function y of x is explicit if it is an equation in which y appears only once, and on its own (usually by starting "y ="). In any other case, the definition of a function y of x is implicit. Implicit differentiation of y with respect to x is a...
  9. J

    Implicit Differentiation: Solving for y' in y^2sin(x)

    Homework Statement Differentiate using implicit differentiation y^2sin(x) Homework Equations I know you need the chain rule and the product rule to solve this The Attempt at a Solution So, it would be: 2yy' + y^2cos(x) Is that correct?
  10. J

    How Do You Apply Implicit Differentiation Correctly?

    Homework Statement Find the derivative of: x+xy=y^2 Homework Equations So I know you have to differentiate it, and it would be: 1+xyy'=2yy' The Attempt at a Solution Moving the terms with y' to one side: 1+xyy'-2yy'=0 xyy'-2yy'=-1 Factoring out y'...
  11. T

    MHB Confusing Implicit Differentiation Problem

    Hi, I have x =(x^2+y^2)^[1/2] I differentiate 1= 1/2 (x^2+y^2)[-1/2] (2x+2yy') So far so good. I try to multiply this out. 1= (2x)/2 (x^2+y^2)[-1/2] + (2yy'/2)(x^2+y^2)[-1/2] I solve for y' y'= 1/{(x (x^2+y^2)[-1/2]} / {y(x^2+y^2)[-1/2] } 1/x (x^2+y^2)[1/2] * 1/y (x^2+y^2)[1/2] The...
  12. B

    Find the Derivative of y2 = 2x + 1How can I find the derivative of y2 = 2x + 1?

    Homework Statement Hello, I missed the class where we were introduced to implicit differentiation so have been catching up this evening. I think I have it, but please could you check my working? Thanks! Find the derivative of y2 = 2x + 1 \frac{d}{dx}([f(x)]^{2}) = \frac{d}{dx}([2x])...
  13. P

    Implicit differentiation and related rates

    Homework Statement The spherical head of a snowperson is melting under the HOT sun at the rate of -160 cc/h (cubic centimetres per hour.) Find the rate at which the radius is changing when the radius r=16. Use cm/h for the units. (The volume of a sphere is given by V= 4π⋅r^3/3.) I have...
  14. C

    MHB Implicit differentiation with exponential function

    find dy/dx: exy+x2+y2= 5 at point (2,0) I'm confused with finding the derivative with respect to x of exy. this is what I did so far for just this part: exy*d(xy)/dx exy*(y+x*dy/dx) do I need to put the parentheses on here? I thought so because that is the part where I used the product rule...
  15. MarkFL

    MHB Duncan G's Questions on Implicit Diff. & Related Rates

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  16. MarkFL

    MHB John's Implicit Diff Q&A: Horiz/Vert Tangents at Yahoo! Answers

    John's question at Yahoo! Answers regarding implicit differentiation & horizontal/vertical tangents Here is the question: I have posted a link there to this topic so the OP can see my work.
  17. A

    Implicit Differentiation to find dy/dx

    Use implicit differentiation to find dy/dx given x^2y+xy^2=4. I have no idea how to approach this problem. My instructor assigned this as homework but has not gone over it at all in class. We have gone over explicit differentiation and I understand this well. I have read the section but it is...
  18. L

    What is the Second Derivative of Implicitly Differentiated Function?

    Homework Statement Determine y'' when 5x^2 + 3y^2 = 4. The Attempt at a Solution So I found the first derivative using the power rule and chain rule, 10x + 6yy' = 0 Which I then solved for y', y' = -10x/6y = -5x/3y Next I found the second derivative using quotient rule...
  19. I

    Jacobian Determinant/ mult. variable implicit differentiation

    Homework Statement Let F: x^2 + y^2 - z^2 + 2xy - 1 = 0 and G: x^3 + y^3 - 5y - 4 = 0. Calculate dz/dx. Note: This is NOT the partial derivative ∂z/∂x. I do not need help in taking the derivative of many polynomials. What I need help in is setting up a Jacobian determinant to evaluate this...
  20. P

    Implicit Differentiation and the Chain Rule

    Hi, I was trying to understand why the chain rule is needed to differentiate expressions implicitly. I began by analyzing the equation used by most websites I visited: e.g. x2+y2 = 10 After a lot of thinking, I got to a reasoning that satisfied me... Here it goes: From my...
  21. D

    A question about implicit differentiation

    So, I understand that implicit differentiation involves derivatives in which x values and y values are mixed up. I've done several implicit differentiation problems a couple sections ago for my math homework, but I pretty just memorized patterns and solved it that way. Now that I'm trying to...
  22. MarkFL

    MHB Josh's question at Yahoo Answers regarding implicit differentiation

    Here is the question: Here is a link to the question: Derive Trig Function? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  23. P

    Implicit Differentiation Question - Stuck

    Homework Statement Use implicit differentiation to find y' given y/(x-y) = x^2+1.Homework Equations The Attempt at a Solution Hi, I'm doing an online course for Calculus 12, and I have been struggling with Implicit Differentiation. I am hoping someone could maybe help me. Thanks. I'm not...
  24. P

    Implicit Differentiation Question - Stuck

    Hi, I'm doing an online course for Calculus 12 and I have been struggling with Implicit Differentiation, hoping someone could maybe help me. Thanks. I'm not positive I'm doing this right but maybe someone can point me in the right direction, this is what I have so far y/(x-y) = x^2+1...
  25. 5

    How to Correctly Apply Implicit Differentiation?

    Homework Statement For the curve x2+3xy+y2 = 5 show that \frac{dy}{dx} =- \frac{2x+3y}{3x+2y} Homework Equations N.A. The Attempt at a Solution 2x + 3xy + 2y\frac{dy}{dx} = 0 3x + 2y \frac{dy}{dx} = -2x+ 3y ∴ \frac{dy}{dx} =- \frac{2x+3y}{3x+2y} have I done this correctly?
  26. T

    Calculus problem- Implicit differentiation

    Homework Statement e^y = x(y-1) answer must be in implicit form Homework Equations The Attempt at a Solution I literally have no idea how to do this problem. I have the answer, but that's it. The answer is dy/dx(e^y) = x(dy/dx) + y - 1
  27. G

    Implicit differentiation in multiple variables

    So, as the title may have given away, I'm trying to figure out implicit differentiation in the multiple variable context. I thought a good practice would be the law of cosines, aka c^2 = a^2 + b^2 - 2abcosθ. So I'm trying to find ∂θ/∂a, ∂θ/db, ∂θ/dc. I tried solving for θ and then taking...
  28. S

    How Do You Isolate dy/dx in Implicit Differentiation?

    Homework Statement Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^2+y^2=(2x^2+2y^2-x)^2 @ (0,(1/2)) Homework Equations The Attempt at a Solution So, I don't have a problem differentiating this. Which I get...
  29. P

    How Do You Isolate dy/dx in Implicit Differentiation?

    Homework Statement Calculate the derivative with respect to x: x/y + y/x = 2yHomework Equations n/a The Attempt at a Solution I end up getting the right answer, but what I want to know is how to isolate dy/dx to one side after implicitly differentiating. I have tried differentiating the LHS...
  30. G

    Implicit Differentiation Problem

    Homework Statement The P(a,b) be a point on the curve √x + √y = 1. Show that the slope of the tangent P is -√b/a Homework Equations ? The Attempt at a Solution Apparently this is an implicit differentiation problem, however we haven't learned or discussed implicit...
  31. J

    Another question about implicit differentiation.

    Say you have x^2+y^2=100. why can't you just solve for y, so y=+- √(100-x^2) then use the chain rule to find the derivative. so y'= +- x/√(100-x^2). Then you can just deduce that y'= -x/y. What is the point of adding all the dy/dx in the equation? Seems like it just complicates it.
  32. M

    Implicit Differentiation for a 2nd Derivative

    Hello! As of right now (10:13 PM), I've tried 9 combinations of points to solve this problem. It's a WebWork-based problem that's due in about an hour in a half. Any help would be very, very appreciated. Homework Statement I was given this equation: ##ln(2y) = 2xy## and was asked to find...
  33. R

    Differentials and Implicit Differentiation

    Homework Statement I'm reviewing physics using Feynman's Lectures, and I'm finding that he frequently uses implicit differentiation in his lessons. This is unfortunate for me because I never got the hang of it beyond the simplest cases. I'm currently going through the proof that the...
  34. D

    Use implicit differentiation to find dy/dx

    Homework Statement Use implicit differentiation to find dy/dx. Homework Equations xey - 10x + 3y = 0 The Attempt at a Solution = [xey + ey(y)'] - (10x)' + (3y)' = 0 = xey + ey(y') - 10 + 3(y') = 0 = y' (ey + 3) = 10 - xey = y' = 10 - xey/ ey + 3y However, my book says the answer: 10 -...
  35. R

    Finding the tangent line using implicit differentiation.

    Homework Statement The equations ##2x^3y+yx^2+t^2=0##, ##x+6+t-1=0## implicitly define a curve $$f(t) = \begin{pmatrix} x(t)\\y(t) \end{pmatrix}$$ that satisfies ##f(1)=\begin{pmatrix} -1\\1 \end{pmatrix}.## Find the tangent line to the curve when ##t=1##. Homework Equations The...
  36. A

    Implicit differentiation help three variables

    Homework Statement I have a question. How in general would one differentiate a composite function like F(x,y,z)=2x^2-yz+xz^2 where x=2sint , y=t^2-t+1 , and z = 3e^-1 ? I want to find the value of dF/dt evaluated at t=0 and I don't know how. Can someone please walk me through this?Homework...
  37. U

    What is the Implicit Differentiation Equation for eysinx=x+xy?

    Equation: eysinx=x+xy I took the derivative of both sides. For the side with eysinx, I used the product rule and chain rule to get: ey*cosx + ey*sinx*y' For the side with x+xy, I used the sum and product rule to get 1+y+xy' So my resulting equation is: ey*cosx + ey*sinx*y'=1+y+xy', which...
  38. J

    Implicit Differentiation Problem

    1. Given that y^{2}-2xy+x^{3}=0, find \frac{dy}{dx} 2. (no relevant equations other than the problem statement) 3. So, I solved it like this, \frac{dy}{dx}y^{2}-2xy+x^{3}=0 2y\frac{dy}{dx}-2+3x^{2}=0 Solving for dy/dx I got... \frac{dy}{dx}=\frac{-3x^{2}+2}{2y}...
  39. J

    Implicit Differentiation, chain rule, and simplifying

    Okay so, I am having trouble figuring out what exactly to do in implicit differentiation and usage of the chain rule. Like, I keep getting the wrong answer somehow. See, from what I understand you have to find the derivative of both sides then use the chain rule or something and then solve for...
  40. J

    Implicit differentiation vs differential equations?

    Hello, I have recently started a little implicit differentiation and I have seen DEs before but I know that I still need to work on my differentiation and integration a little more before I am ready to tackle those. Anyway, I wish to ask, what distinguishes implicit differentiation from a...
  41. D

    Using implicit differentiation: Is this correct?

    Homework Statement I need to use implicit differentiation to find the derivative of y=sin(x+y). Homework Equations The Attempt at a Solution This is what I did: y=sin(x+y) y'=(sin(x+y))' y'=(1+y')(cos(x+y)) (by the chain rule) Now, what do I do? Is this correct...
  42. P

    How Do You Solve Implicit Differentiation for y = sin(xy)?

    1. y = sinxy Homework Equations 3. this was my attempt d/dx = (cosxy)(sinxy(d\dx))+(xy(d/dx) im getting stuck. i don't think I am starting it right. any suggestions.
  43. E

    Implicit differentiation question: can't divide a fraction divided by another

    Homework Statement I'm try to implicitly differentiate the function: xlny+√y=lnx The Attempt at a Solution And I got to the stage where I have: dy/dx = (1/x-lny)/(x/y+1/(2*√y)) which is where I have no idea on how to clean this up. Could someone please explain to me how to simplify a...
  44. C

    More Implicit Differentiation Help

    Homework Statement Find an equation of the tangent line to x^3 + y^3 - 6xy = 0 at the point ((4/3), (8/3)) Homework Equations I got y = -(8/5)x + (24/5). Is this correct? The Attempt at a Solution Lots of algebra involved. Sorry. but I'd rather not type it. I take the derivative of...
  45. C

    How do I find dy/dx for sqrt(xy) = x - 2y using implicit differentiation?

    Homework Statement Find dy / dx for sqrt(xy) = x - 2y. Homework Equations I don't know how to simplify [(xy' + y) / 2sqrt(xy)] = (1 - 2y') to y' = [- y + 2sqrt(xy)] / [x + 4sqrt(xy)]. The Attempt at a Solution I do everything Wolfram Alpha does here...
  46. 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?
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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