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. chwala

    Exploring an Alternative Approach to Implicit Differentiation

    This is a text book example- i noted that we may have a different way of doing it hence my post. Alternative approach (using implicit differentiation); ##\dfrac{x}{y}=t## on substituting on ##y=t^2## we get, ##y^3-x^2=0## ##3y^2\dfrac{dy}{dx}-2x=0## ##\dfrac{dy}{dx}=\dfrac{2x}{3y^2}##...
  2. chwala

    Solve the problem that involves implicit differentiation

    My take; ##6x^2+6y+6x\dfrac{dy}{dx}-6y\dfrac{dy}{dx}=0## ##\dfrac{dy}{dx}=\dfrac{-6x^2-6y}{6x-6y}## ##\dfrac{dy}{dx}=\dfrac{-x^2-y}{x-y}## Now considering the line ##y=x##, for the curve to be parallel to this line then it means that their gradients are the same at the point##(1,1)##...
  3. chwala

    Solve this problem that involves implicit differentiation

    The question and ms guide is pretty much clear to me. I am attempting to use a non-implicit approach. ##\tan y=x, ⇒y=\tan^{-1} x## We know that ##1+ \tan^2 x= \sec^2 x## ##\dfrac{dx}{dy}=sec^2 y## ##\dfrac{dx}{dy}=1+\tan^2 y## ##\dfrac{dy}{dx}=\dfrac{1}{1+x^2}##...
  4. aspiringastronomer

    Struggling in my freshman year of Physics at university

    If Tl;dr I am struggling in Math 171 and Physics 191 and throwing around the idea of declaring a geology major with an astronomy minor because the Physics major "juice is not worth the squeeze" at my age(29) anyone else out there who struggled with Calculus 1 when they first took it?Hello...
  5. mcastillo356

    I Understanding a quote about implicit differentiation

    Hi PF A personal translation of a quote from Spanish "Calculus", by Robert A. Adams: It's about advice on Lebniz's notation1=(sec2⁡y)dydx means dxdx=(sec2⁡y)dydx, I'm quite sure. Why (sec2⁡y)dydx=(1+tan2⁡y)dydx? But I'm also quite sure that the right notation for (sec2⁡y)dydx=(1+tan2⁡y)dydx...
  6. mcastillo356

    Implicit differentiation: why apply the Chain Rule?

    Hi, PF ##y^2=x## is not a function, but it is possible to obtain the slope at any point ##(x,y)## of the equation without previously clearing ##y^2##. It's enough to differentiate respect to ##x## the two members, treat ##y## like a ##x## differentiable function and make use of the Chain Rule...
  7. N

    B Confusion on Implicit Differentiation

    I am confused about implicit differenciation in a few ways. The main confusion is why, in the equation ## x^2 + y^2 = 1 ##, when we are taking the derivative of the left side, ## 2x + 2yy\prime ##, are we adding a ## y\prime ## to the 2y but we aren't adding an ## x\prime ## to the 2x? I also...
  8. qbar

    A Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

    Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
  9. karush

    MHB 2.6.5 Implicit differentiation

    $\tiny{166.2.6.5}$ Find y' $$x^2-4xy+y^2=4$$ dy/dx $$2x-4(y+xy')+2yy'=2x-4y-4xy'+2yy'=0$$ factor $$y'(-4x+2y)=-2x+4y=$$ isolate $$y'=\dfrac{-2x+4y}{-4x+2y} =\dfrac{-x+2y}{-2x+y}$$ typo maybe not sure if sure if factoring out 4 helped
  10. A

    Implicit Differentiation

    Summary:: van der waals I have a pretty good understanding of implicit differentiation. However I'm stuck on a homework problem and could use some help. [P + (an^2)/V^2][V - nb] = nRT a,n,b,R are constants My professor wants me to take the implicit differentiation of P wrt...
  11. EchoRush

    I Questions about implicit differentiation?

    I am new to calculus. I am doing well in my class. I just have a few questions about implicit differentiation. First, why do we call it "implicit" differentiation? Also, when we do it, why when we differentiate a term with a "y" in it, why do we have to multiply it by a dY/dX? What does that...
  12. jisbon

    What is the Solution to Implicit Differentiation Homework with Given Values?

    Homework Statement: Let ##\frac{1}{a}=\frac{1}{b}+\frac{1}{c}## If ##\frac{db}{dt}=0.2## ,## \frac{dc}{dt}=0.3## , Find ##\frac{da}{dt}## when a=80 , b=100 Homework Equations: - Since we are supposed to find ##\frac{da}{dt}##, I can deduce that: ## \frac{da}{dt}...
  13. R

    MHB Implicit differentiation question

    Help! Keep running this and getting different answers, and none are right. 2xy^8 + 7xy = 27 at the point (3,1)
  14. Saracen Rue

    Simple Implicit Differentiation Problem

    Okay so I'm really not sure where I went wrong here; here's how I worked through it: $$\ln\left(y+x\right)=x$$ $$\frac{\frac{dy}{dx}+1}{y+x}=1$$ $$\frac{dy}{dx}+1=y+x$$ If ##\ln\left(y+x\right)=x## then ##y+x=e^x## and ##y=e^x-x## $$\frac{dy}{dx}=y+x-1$$ $$\frac{dy}{dx}=e^x-x+x-1$$...
  15. Zack K

    Value of an implicit derivative

    Homework Statement Find the value of h'(0) if: $$h(x)+xcos(h(x))=x^2+3x+2/π$$ Homework Equations Chain Rule Product Rule The Attempt at a Solution I differentiated both sides, giving h'(x)+cos(h(x))-xh'(x)sin(h(x))=2x+3 Next I factored out and isolated h'(x) giving me...
  16. opus

    Find eqn of Tangent Line to graph- Implicit Differentiation

    Homework Statement Find the equation of the tangent line to the graph of the given equation at the indicated point. ##xy^2+sin(πy)-2x^2=10## at point ##(2,-3)## Homework EquationsThe Attempt at a Solution Please see attached image so you can see my thought process. I think it would make more...
  17. IonizingJai

    Implicit differentiation problem

    Homework Statement If ##x\sqrt{1+y} + y\sqrt{1+x } = 0##, then prove that ##\frac {dy} {dx} = \frac {-1}{(x-1)^2}##. 2.Relevant Equations: $$ \frac {dy} {dx} = - \frac {\left (\frac {\partial f}{\partial x} \right)} {\left( \frac {\partial f} {\partial y} \right)}.$$ 3...
  18. KFSKSS

    I need some help with implicit differentiaiton.

    Hello. My problem is as follows: Suppose x^4+y^2+y-3=0. a) Compute dy/dx by implicit differentiation. b) What is dy/dx when x=1 and y=1? c) Solve for y in terms of x (by the quadratic formula) and compute dy/dx directly. Compare with your answer in part a). I solved a) and b). a)=-4x^3/2y+1, and...
  19. H

    MHB Implicit Differentiation to find equation of a tangent line

    I need urgent help. I have this question: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \begin{equation} {x}^{2/3}+{y}^{2/3}=4 \\ \left(-3\sqrt{3}, 1\right)\end{equation} (astroid) x^{\frac{2}{3}}+y^{\frac{2}{3}}=4 My answer is...
  20. A

    Implementing symmetry boundary condition for the diffusion equation

    The following lines of codes implements 1D diffusion equation on 10 m long rod with fixed temperature at right boundary and right boundary temperature varying with time. xsize = 10; % Model size, m xnum = 10; % Number of nodes xstp =...
  21. M

    Finding dy/dx for a circle

    Homework Statement Hello I have this circle with the equation : [/B] (x-a)^2+(y-b)^2=r^2 I want to find dy/dx for it 2. Homework Equations (x-a)^2+(y-b)^2=r^2 The Attempt at a Solution I am looking on the internet and it appears that I should use what is called "Implicit differentiation"...
  22. Blockade

    B Is dy/dx of x2+y2 = 50 the same as dy/dx of y = sqrt(50 - x2)?

    For implicit differentiation, is dy/dx of x2+y2 = 50 the same as y2 = 50 - x2 ? From what I can take it, it'd be a no since. For x2+y2 = 50, d/dx (x2+y2) = d/dx (50) --- will eventually be ---> dy/dx = -x/y Where, y2 = 50 - x2 y = sqrt(50 - x2) dy/dx = .5(-x2+50)-.5*(-2x)
  23. P

    MHB Edin's question via email about implicit differentiation

    (a) Differentiate both sides of the equation with respect to x: $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ y^3 + y + x\,y^2 \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \left[ 10 + 4\sin{(x)} \right] \\ 3\,y^2\,\frac{\mathrm{d}y}{\mathrm{d}x} +...
  24. Debaa

    B Implicit differentiation or just explicit?

    How do I figure whether to do implicit differentiation or just explicit?? Thanks for the answer.
  25. Schaus

    Find the equation of the tangent line of the curve

    Homework Statement Find the equation of the tangent line to the curve ##\ xy^2 + \frac 2 y = 4## at the point (2,1). Answer says ##\ y-1 = -\frac 1 2(x-2)## And with implicit differentiation I should have gotten ##\frac {dy} {dx}= -\frac {y^2} {2xy-\frac {2} {y^2}}## Homework Equations ##\...
  26. FritoTaco

    Implicit Differentiation

    Homework Statement \dfrac{x^2}{x+y}=y^2+8 Homework Equations Quotient Rule: \dfrac{g(x)\cdot f'(x)-g'(x)\cdot f(x)}{(g(x))^2} Product Rule: f(x)\cdot g'(x)+g(x)\cdot f'(x) The Attempt at a Solution \dfrac{(x+y\cdot\dfrac{dy}{dx})(2x)-(1\cdot\dfrac{dy}{dx})(x^2)}{(x+y\cdot...
  27. J

    Implicit differentiation of many variables

    Homework Statement For the given function z to demonstrate the equality: [/B]As you see I show the solution provided by the book, but I have some questions on this. I don't understand how the partial derivative of z respect to x or y has been calculated. Do you think this is correct? I...
  28. I

    I Implicit differentiation

    Hey, I found a thread about part of what I'm trying to ask long ago: https://www.physicsforums.com/threads/implicit-differentiation.178328/ Basically, I noticed that if you multiply by x or by y in an equation before implicitly deriving, you get two different answers. Unfortunately their whole...
  29. R

    Implicit Differentiation Question

    Homework Statement I am told to find dy/dx by implicit differentiation where: e^(x^2 * y) = x + y Homework Equations The above equation and the ln of it.The Attempt at a Solution e^(x^2 * y) = x + y (x^2 * y)ln(e) = ln(x+y) x^2 * y = ln(x+y) x^2(dy/dx) + y(2x) = 1/(x+y) * (1 + dy/dx)...
  30. U

    MHB Find the derivative using implicit differentiation (with inverse trig functions)

    Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...
  31. Q

    Second Derivative (Implicit Differentiation)

    Homework Statement Find y'' Homework Equations 9x^2 +y^2 = 9 The Attempt at a Solution y' 18x+2y(y')=0 y'=-18x/2y y'=9x/y For the second derivative, I get the correct answer (same as the book) up until the very last step. Here's where I'm left at: -9( (-9x^2 - y^2) / y^3 ) The book then...
  32. T

    Implicit Differentiation Question

    << Mentor Note -- thread moved from the technical math forums at OP request, so no Homework Help Template is shown >> x2y + xy2 = 6 I know we use the chain rule from here, so wouldn't that be: (d/dx)(x2y + xy2) = (d/dx)(6) so using the chain rule of g'(x)f'(g(x) and the d/dx canceling out on...
  33. Q

    Implicit differentiation (beginner)

    Homework Statement Find y' ... X^2+y^2=25I understand (I think) implicit differentiation, but there is one issue which hangs me up. I've done this before and this is just a refresher as my last calculus course was four years ago. From what I understand, 2x+2y(y')=0 But why isn't it...
  34. M

    I Implicit differentiation

    First of all thanks for the help, i have a problem finding a good explanation of de ecuation (d/dx)f=(∂f/∂x)+(∂f/∂y)*(dy/dx) could anyone write me a good explanation of this ecuation? thanks for the help
  35. A

    B (ASK) Implicit Differentiation

    What is ##\frac{d}{dx}(\frac{x}{y^2})##? Please tell me is it correct or not: ##\frac{d}{dx}(\frac{x}{y^2}) = \frac{[\frac{d}{dx}(x)] ⋅ (y^2) - (x) ⋅ [\frac{d}{dx} (y^2)]}{(y^2)^2}## ## = \frac{(x) ⋅ (y^2) - (x) ⋅ (\frac{d}{dy} (y^2)) ⋅ \frac{dy}{dx}}{y^4}## ##= \frac{xy^2 -...
  36. P

    MHB Kamal's Questions via email about Implicit Differentiation

    Since we have this relationship between x and y, as the two sides are equal, so are their derivatives. We just have to remember that as y is a function of x, any function of y is also a function of x, with the inner function "y" composed inside whatever is being told to do to the y. So to...
  37. P

    MHB Effie's question via email about Implicit Differentiation

    To perform implicit differentiation we must make use of the chain rule. Basically if you have a function composed in another function, its derivative is the product of the inner function's derivative and the outer function's derivative. All other rules (such as the sum rule, the product rule...
  38. R

    MHB Implicit differentiation

    Hello! Can someone help me with the process of solving \sqrt{x}+\sqrt{y}=5 on point (4,9)? With implicit, I differntiated both sides and ended up with 1/2x^-1/2+1/2y^-1/2\d{y}{x}=0 and I tried to isolate the dy/dx, but how do I get rid of the others? And with explicit, I isolated y to one side...
  39. I

    Basic implicit differentiation question

    So it has been quite a few years since I learned about implicit differentiation so the content is a bit rusty in my mind. x=rcos(θ) How do you find dx/dt? I know the answer but I am trying to figure out why. I mean dx/dt can be written as (dx/dθ)*(dθ/dt) so why is the answer not just...
  40. R

    MHB Help on Related Rates implicit differentiation

    Hi! I recently came upon this problem : the height of a right angled triangle is increasing at a rate of 5cm/min while the area is constant. How fast must the base be decreasing at the moment when the height is 5 times the base? I drew a picture of the triangle, labelled the height (h) and...
  41. G

    MHB How do I differentiate $\cos(x+y)$?

    If $y^2+\cos(x+y) = 1$ find $\frac{dy}{dx}$. How do I differentiate $\cos(x+y)$ bit?
  42. X

    Implicit Differentiation z=f(x/y) meaning

    Mod note: Moved from the Homework section 1. Homework Statement This might seem like a stupid question but I'm unsure what z= ƒ(x/y) means? I'm not sure how I would find ∂z/∂x , ∂z/∂y just from this statement either. Thank you Homework EquationsThe Attempt at a Solution
  43. iwantcalculus

    Implicit differentiation question with inverse trig

    Homework Statement Homework Equations The Attempt at a Solution Note: by real solution I mean the correct implicit derivative, not an actual real solution... Please help![/B]
  44. B

    How is implicit differentiation performed in calculus?

    Folks, Differentiate implicitly \phi(x,y)=0 I get: wrt to x \phi_x+\phi_y \frac{dy}{dx} and wrt to y \phi_y+\phi_x \frac{dx}{dy} however I don't know how this is derived \phi_x dx+\phi_y dy=0
  45. B

    MHB Yes, that makes sense! Thank you for explaining it to me.

    Hi Folks, It is been given that differentiation of \phi(x,y)=0 is \phi_{x} dx+ \phi_{y} dy=0 however I arrive at \phi_{x} dx/dy+ \phi_{y} dy/dx=0 via the chain rule. Where \phi_{x}=d \phi/dx etc What am I doing wrong? Thanks
  46. K

    Implicit Differentiation

    I have an equation: r^2 = x^2 So I found out dr/dx = x/r. But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3. Can anyone help? My working out: r^2 - x^2 = 0 r^2 = x^2. Assume r is a function of x. rr' = x (first derivative...
  47. funlord

    Implicit Differentiation: two different answers

    Homework Statement with answers given: Homework Equations use implicit differentiation The Attempt at a Solution I always get this answer but not the second one PLs explain the second answer for I am very desperate. Thank You
  48. karush

    MHB Implicit differentiation

    $$6x-\sqrt{2xy}+xy^3 ={y}^{2}$$ $$6-?+3x{y}^{2}{y'}^{}+{y}^{3}=2y{y'}^{}$$ Got stumped on this one answer was complicated...
  49. Drakkith

    Implicit Differentiation: Differentiating in Terms of X

    I'm having some trouble with the terminology used in calculus. My book states: "Fortunately we don't need to solve an equation for Y in terms of X in order to find the derivative of Y. Instead we can use the method of implicit differentiation. This consists of differentiating both sides of the...
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