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. E

    Using implicit differentiation to differentiate log_a (x)

    Could someone please make sure I'm doing this right. I want to find the derivative of the logarithm to the base a of x, using implicit differentiation. Let y = \log_{a} x a^y = x \frac{d}{dx} (a^y) = 1 (implicit differentiation) \frac{d}{dx} (e^{\ln a})^y = 1 \frac{d}{dx} (e^{(\ln a)y}) = 1...
  2. N

    Implicit Differentiation - Second Derivative

    Were assigned questions regarding implicit differentiation and the second derivative but did not receive a formal lesson so I need some explanations. Example: Find the second derivative x^3 + y^3 = 1 I found this solution on the internet and the answer matches the one in the textbook...
  3. B

    Implicit Differentiation in Multivariable Calculus

    I need to compute the partials of z with respect to x and y of: xy + z + 3xz^5 = 4 at (1,0). I already showed that the equation is solvable for z as a function of (x,y) near (1,0,1) with the special implicit function theorem, but that's the easy part. Could someone explain to me how to begin...
  4. V

    Implicit Differentiation Problem

    Consider the curve given by X^2+4y^2=7+3xy a) show that dy/dx=3y-2x/8y-3x b) show that there is a point P with x-cooridnate 3 at which the line tangent to the curve at P is horizontal. Find the y-cooridnate of P. c)find the value of d^2y/dx^2 at the point P found in part (b). Does the curve...
  5. Q

    Finding the Derivative of a Function with Implicit Differentiation

    Assume that y is a function of x . Find y' = dy/dx for (x^3+y^3)^20 when i solved this i got y'= (20(x^3+y^3)^19 * 3x^2)/(-3y^2) is this correct or am i missing something?
  6. T

    Implicit Differentiation of Multivariable Functions

    x^2+y^2+r^2-2s=13=0 x^3-y^3-r^3+3s+59=0 How do I find the partial derivatives of x(r,s) or y(r,s) implicitly? I tried implicit differentiation and I got 2 different answers for either. Can someone show me any of the 4 derivatives step-by-step?
  7. G

    Another implicit differentiation Problem

    The question I'm having trouble with is as follows: Given that siny = 2sinx show that: a) (dy/dx)^2 = 1+3sec^2(y), by differentiating this equation with respect to x show that b)d^2y/dx^2 = 3sec^2ytany and hence that c) coty(d^2y/dx^2) - (dy/dx)^2 + 1 = 0 Part (c) is straight forward and...
  8. D

    Finding the Second Derivative Using Implicit Differentiation

    Can someone check my answer (I am trying to find the second derivative) for any mistakes? I have looked it over many times, and I've realized that my second derivative is not correct, but I cannot figure out why. Thank you. \sqrt{x} + \sqrt{y} = 1 \frac{1}{2\sqrt{x}} +...
  9. M

    Implicit differentiation problem

    hello everyone I'm stuck! anyone have any ideas? I'm suppose to find dz/dx and dz/dy with implicit differentation. This is calc III! http://img221.imageshack.us/img221/4000/lastscan4ou.jpg
  10. 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)}
  11. 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...
  12. 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}...
  13. I

    Why Does Implicit Differentiation Give Contradictory Results for u_z?

    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...
  14. 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...
  15. C

    Finding the Derivative of Implicit Functions

    Use Implicit Differenciation to find y' f tan(x^2 + y^2) = sec(xy)
  16. 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
  17. 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?
  18. 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...
  19. 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...
  20. 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...
  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. 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'=...
  26. 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...
  27. 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...
  28. 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...
  29. 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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