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. F

    Implicit differentiation dy/dx

    Homework Statement Determine dy/dx xy^-2 = 1 Homework Equations The Attempt at a Solution Here's what I tried: y^-2 - 2xy^-3(dy/dx) = 1 dy/dx = (y - y^3)/2x The answer in the book is y/2x. How do you do it?
  2. K

    How Do You Solve for d²y/dx² in Implicit Differentiation?

    Homework Statement If siny=2sinx and (dy/dx)^2=1+3sec^2(y) show that: by differentiating 1+3sec^2(y) with respect to x, d^2y/dx^2=3sec^2(y)tan(y) Homework Equations The Attempt at a Solution
  3. P

    How to find the rate of decrease of water height in a tank

    Homework Statement Water flows from a tank of constant cross-sectional area 54 ft2 through an orifice of constant cross-sectional area 1.7 ft2 located at the bottom of the tank. Initially the height of the water in the tank was 20 and its height t sec later is given by the following...
  4. B

    Second Derivative using Implicit Differentiation

    Homework Statement Find y''(x) of the parametric equation 9x^2+y^2=9 using implicit differentiation. Homework Equations I already came up with y'(x) = -9x/y The Attempt at a Solution Here is what I have for y''(x) so far y''(x) = d/dx (-9xy^-1) =-9(d/dx)(xy^-1)...
  5. B

    Implicit Differentiation: two different answers found. Please explain.

    Homework Statement Could anyone explain that I got two different answers for this question: find dy/dx of \frac{x}{x+y}-\frac{y}{x}=4 Homework Equations The Attempt at a Solution 1. using quotient rule: \frac{x+y-(1+dy/dx)x}{(x+y)^{2}}-\frac{x\frac{dy}{dx}-y}{x^{2}}=0...
  6. B

    Implicit differentiation: two answers resulted

    Could anyone explain that I got two different answers for this question: find dy/dx of \frac{x}{x+y}-\frac{y}{x}=4. 1. using quotient rule: \frac{x+y-(1+dy/dx)x}{(x+y)^{2}}-\frac{x\frac{dy}{dx}-y}{x^{2}}=0...
  7. B

    Finding Points of Tangency for a Given Tangent Line Equation

    The function of a line is y^2 + x^3 = 9. I calculated the slope of its tangent to be -3x^2/2y. The question asked us to find a point(s) so that the equation of its tangent line is y + 6x = 13. So it's slope must be -6 at that point. I got (2,1) as a point. Are there more than one, or is that...
  8. W

    Find Tangent Line to Cardioid at (0, 0.5) Using Implicit Diff.

    Homework Statement Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5). x2 + y2 = (2x2 + 2y2 - x)2Homework Equations Derivative rules point slope formula The Attempt at a Solution I got y' = [16x3-4x2+16xy2-4y2-4y2-8x2+2x] / [2y -...
  9. A

    Implicit Differentiation Clarification

    Homework Statement 2x^{2} - 3y^{2} = 4 Homework Equations We say that y is an implicit function of x if we are given an equation: \sigma(x,y) = \tau(x,y) Then to differentiate we do: \frac {d(\sigma(x,y))} {dx} = \frac {d(\tau(x,y))} {dx} The Attempt at a Solution...
  10. S

    How do I use implicit differentiation to find dy/dx in this given equation?

    Homework Statement Use implicit differentiation to find dy/dx if y - sin(xy) = x^2. What I've got is dy/dx y - cos(xy)(y+x dy/dx) = 2x I don't know what I did and I don't know where to go from here.
  11. L

    How Do You Solve This Implicit Differentiation Problem?

    Homework Statement find \frac{\mathrm{d}y}{\mathrm{d}x} where y is defined implicitly as a function of x Homework Equations x\sin(xy)=x The Attempt at a Solution x(\cos(xy)(x\frac{\mathrm{d}y}{\mathrm{d}x}+y))+\sin(xy)=1...
  12. J

    Proof using implicit differentiation

    Prove that if P(a,b) is a point on the rotated ellipse C (whose equation is x^2 -xy +y^2=4 ), then so is Q(-a,-b), and that the tangent lines to C at P and Q are parallel. The equation of the line joining P and Q is y - b = m(x - a), where m= \frac{b-(-b)}{a-(-a)} = \frac{b}{a}, then the...
  13. Salazar

    Swimming Pools and Related Rates along with Implicit Differentiation

    Homework Statement A swimming pool is 40 feet long, 20 feet wide, 4 feet deep at the shallow end, and 9 feet deep at the deep end. Water is being pumped into the pool at 10 cubic feet per minute. a. When the water is 3 feet deep at the shallow end, at what rate is the water level rising? b...
  14. Salazar

    Air Traffic Controller and Related Rates along with Implicit Differentiation

    Homework Statement "An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 120 miles from the point and is moving at 400 miles per hour. The other plane is 160 miles from the point and has a speed of...
  15. T

    Finding Points on a Curve with Tangent Line Slope -1

    Homework Statement Find all the points on the curve x^{2}y^{2}+xy=2 where the slope of the tangent line is -1. The Attempt at a Solution I differentiated both sides of the equation and got: \frac{dy}{dx}=\frac{-2xy^{2}-y}{x^{2}2y+x} I know that \frac{dy}{dx}=-1, but if I substitute...
  16. L

    Implicit Differentiation w/ Composite Function

    Homework Statement Given the equation y= f(x) , at a certain point the slope of the curve is 1/2 and the x-coordinate decreases at 3 units/s. At that point, how fast is the y-coordinate of the object changing? The Attempt at a Solution Dy/dx = f ' (x) dx/dt Would that be...
  17. J

    Implicit Differentiation: Finding Tangent Line and Second Derivative

    Homework Statement Consider: x^3+y^3+2xy=4, y=1 when x=1 a.) Find the equation of the tangent line to the curve when x=1. b.) Find y'' at x=1. c.) Is the graph of y=f(x) concave up or concave down near x=1? Homework Equations Any derivative rules... The Attempt at a Solution For Part a...
  18. D

    Understanding Implicit Differentiation: Solving for Second Order Derivatives

    Find d^2/dx^2(3y^2+8y=3x) I managed to get dy/dx = 3 / (6y + 8) but I have no clue where to go from here. According to WolfRamAlpha, the answer is -27/(4(16 + 9x)(4 + 3y)), but since dy/dx doesn't have any x value in it, I don't see how the derivative of it would. I've played around...
  19. D

    Implicit differentiation problem.

    Homework Statement Find y´(x) for x + y = 1/x^2 + 8/y^2 Homework Equations The Attempt at a Solution Rewrite the eq. as x + y = x^-2 + 8y^-2 differentiate. 1 + y´(x) = (-2x^-3) + (-16y^-3)(y´(x)) Rearrange 1 + (2x^-3) = (-16y^-3)(y´(x)) - (y´(x)) (1 + (2x^-3))/(16y^-3) =...
  20. B

    Find y'' by implicit differentiation

    Homework Statement Find y'' by implicit differentiation. Homework Equations The Attempt at a Solution I get to this point in the problem, which is I solved for y'. But then when I attempt to take y'', in other words take the derivative of my answer for y', I...
  21. M

    Implicit differentiation find dy/dx and d2y/dx2

    2xy=3x-y^2 find dy/dx and d2y/dx2 I just want to make sure my answer is right and simplified I got dy/dx= (3-2y)/ (2x+2y) Now d2y/dx2 took some time but this is what i got: (-12x+2x2y+9+4yx-14y-4y^2) / ((x+y)(2x+2y)^2)
  22. T

    Implicit differentiation on x^3 + y^3 = 4xy + 1

    1. use implicit differentiation to evaluate y(prime) for x^3+y^3=4xy+1 at the point (2,1)
  23. A

    Solve using implicit differentiation 6 y+8 x=\sin(xy^2)

    so to solve this i differentiated each part and got 6dy/dx + 8 = cos(xy^2)(y^2*x2ydy/dx) then i divided both sides by cos(xy^2) then serpatated the 6dy/dx + 8 and put them both over cos(xy^2) then i took out a yprime and ended up with -8/(6-2y^3xcos(xy^2)) as an answer but its wrong...
  24. V

    Doubt about Implicit differentiation

    Hi all, I was reading a paper in which implicit differentiation was used as follows x \in R, \lambda \in R Given G(x,\lambda) = 0 \frac{\partial G(x,\lambda)}{\partial x} \frac{\partial x}{\partial \lambda} + \frac{\partial G(x,\lambda)}{\partial \lambda} = 0 My doubt is...
  25. L

    Implicit Differentiation and product rule

    Hi, So, I am reviewing Cal III, and I have come across something that I do not understand regarding implicit differentiation with partial derivatives: x^3 + y^3 + z^3 + 6xyz = 1 implicit differentiation of z with respect to x: 3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0 *notive the...
  26. K

    Finding Partial Derivatives with Implicit Differentiation

    Homework Statement Use implicit differentiation to find ∂z/∂x and ∂z/∂y yz = ln(x + z) The Attempt at a Solution I came up with (x+2)/(x+2)(1-xy-yz) Could someone please help me solve this. I know to treat y as a constant and to multiply all the derivatives of z by ∂z/∂x
  27. H

    Help: Implicit differentiation with initial values

    Homework Statement 2*y + sin(y) = x^4 + 4(x)^3 + (2(Pi) - 5), show that dy/dx = 16, when x = 1. Homework Equations The Attempt at a Solution So I implicitly differentiated it to be dy/dx(2 + cos(y)) = 4(x)^3 + 12(x)^2, and I end up with dy/dx = 16 / (2 + cos (y)) which means that...
  28. M

    Implicit Differentiation: Finding Derivatives of an Implicit Function

    Homework Statement Assume that the following equation define the implicit function y=(x). Find the its derivative: x2 + 2xy - y2 = a2 y'=? y''=? Homework Equations \frac{dy}{dx} = -\frac{F_x}{F_y} The Attempt at a Solution so for the first derivative I express that equation as...
  29. K

    Implicit Differentiation and understanding the question?

    Hi, I am working on my differential equations excercises and I encountered 2 problems. First one is, I just wanted to check if I did this implicit differenriation right Homework Statement t^{2} \bullet y +y^{2} = C where is is a constant The Attempt at a Solution My solution is y...
  30. M

    Explicit Equations for Implicit Set at (0,0): First Partial Derivatives

    Homework Statement Show that the set defined by the equations x + y + z + w = sin(xyzw) x - y + z + w^2 = cos(xyzw) - 1 can be described explicitly by equation of the form (z, w) = f(x, y) near the point (0,0,0,0); find the first partial derivatives of f(x,y) at the point (0,0)...
  31. P

    What Is the Correct Approach to Solve These Implicit Differentiation Problems?

    Homework Statement 8x^2-10xy+3y^2=26 2. The attempt at a solution (8)(2x)-(-10x)y'+(y)(-10)+(3)(2y)y'=0 16x+10x(y')-10y+6y(y')=0 y'(10x+6y)+16x-10y=0 y'(10x+6y)=10y-16x y'=(10y-16x)/(10x+6y) y'=(5y-8x)/(5x+3y) I know I'm doing something wrong but I can't see it for...
  32. S

    How do I find f'(1) using implicit differentiation?

    "implicit differentiation" if [x][3] * f(x) + [(f(x))][3] + f([x][3]) = 3 and f(1)= 2 find f'(1) NEED HELP REVIEW QUESTION FROM EXAM REVIEW DONT KNOW WHAT TO DO
  33. M

    A run of the mill Implicit Differentiation

    Homework Statement Use Implicit Differentiation to find y' of the equation 5x^2+ 3xy+y^2=152. The attempt at a solution y'= (-10x-5y)/3x I would like to know if I did this right. I am not very confident in my math sometimes that why I came here. If i did this wrong will you please steer me...
  34. L

    Solve by Implicit Differentiation or Partial Differentiation?

    Homework Statement I've got a question more with the structure of how this problem is presented: If x^(sin y) = y^(cos x) Find \frac{dx}{dy}(\frac{pi}{4},\frac{pi}{4}) Homework Equations We have been taught to solve by implicit...
  35. O

    Implicit Differentiation: Find Y

    Homework Statement Find y' Homework Equations The Attempt at a Solution My solution: it's correct?
  36. P

    Implicit Differentiation - Tangent Line & Horizontal Tangents

    Homework Statement Find an equation of the tangent line to this curve at the point (1, -2). Homework Equations The Attempt at a Solution 2y' = 3x^2+6x y' = 3x^2+6x y'=3/2x^2+3x y+2=3(x-1) y+2=3x-3 y=3x-5
  37. P

    Implicit Differentiation with Sin

    Homework Statement Homework Equations The Attempt at a Solution
  38. S

    What is the Second Derivative Using Implicit Differentiation?

    Use Implicit Differentiation to find y" if xy + y - x = 1 so far i got 1y + dy/dx - dx/dx = 1/dx then i did y + y' - 1 = 0 y' = 1-y i don't understand how to get the y" . i don't think i even have y' done right!
  39. H

    Implicit Differentiation Question

    Homework Statement Prove that \frac{dy}{dx}=\frac{y}{x} for \sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=10 x is not equal to y which is not equal to 0 The Attempt at a Solution Tried all the normal methods but none seem to work...anyone have any ideas?
  40. P

    How do I solve for y' in implicit differentiation problems?

    Homework Statement Find derivative of y with respect to x. Sin(xy) = Sinx Siny Homework Equations The Attempt at a Solution Use chain rule (product rule for inner function) to differentiate the left. Use product rule to differentiate the right and I get the following...
  41. T

    Unraveling the Implicit Differentiation of y=vx

    Homework Statement Consider y = 2a + ax find dy/dx dy/dx = a That is right is it not, as a is treated merly as a constantNow consider this question: Use the substitution y = vx to transform the equation: dy/dx = (4x+y)(x+y)/x² into x(dv/dx) = (2+v)² According to the mark scheme they...
  42. S

    Integration after implicit differentiation

    Homework Statement What is the integral of: y + x \frac{dy}{dx} The Attempt at a Solution I know that it is xy, after implicit differentiation. However, I cannot get it without prior knowledge of implicit differentiation.
  43. M

    Implicit differentiation - messed up somewhere

    Homework Statement find d^2y/dx^2 at (1,1) for: x^2y + x^2 -y^2 = 1 Homework Equations none The Attempt at a Solution i worked it all out but the answer I am getting is not an option. could someone show me where i made a mistake? I am not asking you to do the problem for me...
  44. A

    Help with 1 quick implicit differentiation problem

    Homework Statement what is the implicit differentiation ? Homework Equations 2sin(x)cos(y)=1 The Attempt at a Solution d/dx[2sin(x)cos(y)]= d/dx[1] 2cos(x)*-sin(y)*dy/dx=0 I haev a bad feeling i did this wrong...
  45. U

    Implicit differentiation homework

    hi i couldn't solve this question i think there is a mistake in it anyone can check it please ? http://img296.imageshack.us/img296/3716/13055173zr3.png
  46. K

    Implicit Differentiation and coordinates

    Homework Statement Find the coordinates of the point in the first quadrant at which the tangent line to the curve x3-xy+y3=0 is parallel to the x-axis. SO: x= + y= + mtan=0 Homework Equations \frac{dy}{dx}=m_{tan} The Attempt at a Solution \frac{dy}{dx}=\frac{y-3x^{2}}{3y^{2}-x}=0 After I...
  47. N

    2 Problems. 1 regarding implicit differentiation

    1) A particle is moving along the curve y=3sqrt{3x+3}. As the particle passes through the point (2, 9), its x-coordinate increases at a rate of 2 units per second. Find the rate of change of the distance from the particle to the origin at this instant. 2) Use implicit differentiation to find...
  48. N

    How do I find the slope of a tangent line using implicit differentiation?

    Homework Statement Use implicit differentiation to find the slope of the tangent line to the curve 4x^2-3xy+1y^3=26 at the point (3,2) The Attempt at a Solution I attempted the problem and i came up with dy/dx= (-8x+4)/(3y^2) which is wrong. Need some help with this.
  49. D

    Implicit Differentiation w/ trig functions check

    Homework Statement Determine dy/dx when y*sin(x2)=5 The Attempt at a Solution y*2xcos(x2) dx/dx + sin(x2)*1 dy/dx = 0 2xy cos(x2)*dy/dx = -sin(x2) dy/dx = -sin(x2) / 2xy cos(x2) dy/dx = -2xy tan(x2)
  50. D

    Check request: Implicit Differentiation

    Homework Statement Find the value of dy/dx at x=2 when x2+xy=5 The Attempt at a Solution To find y: 22+2y=5 y=1/2 To find dy/dx: 2x(dx/dx)+x(dy/dx)+y(dx/dx)=0 2x+x(dy/dx)+y=0 (dy/dx)=(-2x-y) / x Plug in y and x: dy/dx=-2(2)-(1/2) / 2 dy/dx=-9/4 when x=2
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