# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### How do you properly apply the chain rule in implicit differentiation?

The working I've tried is in the attachment.
9. ### 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...
10. ### 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
11. ### 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...
12. ### 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...
13. ### 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}...
14. ### 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)
15. ### 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## $$\frac{dy}{dx}=y+x-1$$ $$\frac{dy}{dx}=e^x-x+x-1$$...
16. ### 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...
17. ### 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...
18. ### 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...
19. ### 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...
20. ### 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. $${x}^{2/3}+{y}^{2/3}=4 \\ \left(-3\sqrt{3}, 1\right)$$ (astroid) x^{\frac{2}{3}}+y^{\frac{2}{3}}=4 My answer is...
21. ### 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 =...
22. ### 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"...
23. ### 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)

43. ### 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
44. ### 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]
45. ### 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
46. ### 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
47. ### 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...
48. ### 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
49. ### 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...
50. ### 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...