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Questions tagged [implicit-differentiation]

For questions on finding and evaluating derivatives when a function is defined implicitly.

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Why does implicit derivative change after division? (For $\sec(x)+\tan(y) = \sec(x)*\tan(y))$)

If I take the implicit derivative without dividing, I get the solution $\frac{dy}{dx} = (\sec(x) \tan(x) \cos(y)) \frac{\cos(y)-\sin(y))}{\sec(x)-1}$. If I divide both sides by $\sec(x) \tan(y)$, I ...
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A particle moves along the curve… Find component of acceleration [on hold]

differentiation of vector to find components of velocity and accelration (https://i.stack.imgur.com/djAtB.jpg)
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9 views

Implicit differentiation applied to $ z=\frac{1}{y}(f(ax+y)+g(ax-y)). $

I'm trying show that $$\frac{\partial^2z}{\partial x^2}=\frac{a^2}{y^2}\frac{\partial}{\partial y}( y^2\frac{\partial z}{\partial y})$$knowing that: $$ z=\frac{1}{y}(f(ax+y)+g(ax-y)). $$ I know that,...
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27 views

Can implicit differentiation be used when differentiating with respect to the dependent variable?

EDIT: I forgot to add that I desperately need clarification on this as I will be sitting an exam shortly. Sorry in advance if this does not comply with guidelines -- I'm still new to this site. I'll ...
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1answer
47 views

Implicit Function Theorem with $x^2+y^2+z^2=\psi(ax+by+cz)$

Given the equation $x^2+y^2+z^2=\psi(ax+by+cz)$, with $a,b,c\in\mathbb{R},\ c\neq 0$, and $\psi:\mathbb{R}\to\mathbb{R}$ that satisfies $\psi\in C^2,\ \psi(0)=0,\ \psi'(0)\neq0$, prove that in a ...
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1answer
24 views

Second derivative of multivariable implicit function

Find $d^2z$ of the following function $$\frac{x}{z}= \ln{\frac{z}{y}}+1$$ Finding the total derivative of the left and right side we arrive at the expression: $$\frac{zdx-xdz}{z^2}=\frac{y}{z} \...
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1answer
48 views

implicit differentiation related rates, rate of an angle

A particle moves along the graph of $y=x^2$ over the plane xy at a constant velocity of 10cm/s. Let θ denote the angle between the x-axis and the line that goes from the origin to $P(x,x^2)$. find ...
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1answer
65 views

What can be said about $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ solutions?

There was an unanswered question 4 years ago. OP asked for a solution of ODE $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ The comment to the original question proposes an implicit solution, $2\log y + 2\...
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1answer
44 views

Differentiation of inverse function

I know that first order differentiation of inverse of a function $f (x)$ is reciprocal of $f'(f^-1(x)) $. But I'm unable to evaluate the integration given in the question.
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How do i find the acceleration of a particle in circular motion?

How do i find the acceleration of a particle in circular motion? I can find out the component $r\alpha$ using differentiation, but how can i find both rectangular components of the acceleration vector?...
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$F(x - az, y - bz) = 0$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$

Show that the function $z = z(x, y)$ determined implicitly by the equation $F(x - az, y - bz) = 0$, in which $F$ is a differentiable function, satisfies the equation $$ a\frac{\partial z}{\...
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1answer
26 views

Implicit multivariable derivative in polar coordinates

Determine the first and second orders derivatives of $y = f(x)$ determined implicity by $$ \ln\sqrt{x^2 + y^2} = \alpha~\arctan\frac{y}{x} $$ Now, notice that in polar coordinates, the ...
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3answers
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Prove that the second order implicit multivariable derivative equals $0$

Let $ y $ be determinaded implicitly as a function of $ x $ by the equation $ x^2 + y^2 + 2axy = $ 0, with $ a > 1 $. Prove that $ \frac{d^2y}{dx^2} = 0 $. My failed attempt $$ \frac{dy}{dx} = - \...
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Implicit differentiation from backwards euler in python.

I have a problem with my backward euler formula which I need to solve in implicit way in python. h = 0.1 r = 10 So my backwards euler is d[i+1] = d[i]*(1(1+h*r)) ...
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How to derive this complex differentiation?

In the finite-deformation theory, the elastic Cauchy-Green strain $\mathbf{E}_e$ is defined as $\mathbf{E}_e=\frac{1}{2}(\mathbf{F}_e^T \mathbf{F}_e-\mathbf{I})$, where the superscript $T$ denotes ...
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An expression for computing second order partial derivatives of an implicitely defined function

Let $\Phi(x,y)=0$ be an implicit function s.t. $\Phi:\mathbb{R}^n\times \mathbb{R}^k\rightarrow \mathbb{R}^n$ and $\det\left(\frac{\partial \Phi}{\partial x}(x_0,y_0)\right)\neq 0$. This means that ...
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Fly traveling a through a point, along curve of intersection

The temperature in 3-space is given by: $$ T(x,y,z)= \frac12(2x^2+5y^2+4z^2) $$ At time $$t = 0,$$ a fly passes through the point $$(\sqrt15,\sqrt10,5),$$ flying along the curve of intersection of ...
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1answer
49 views

Implicit Differentiation - Different Approaches 2

Given is the function $F(x,y,z)=x^2+y^3-z$. Determine the Jacobian matrix $Dz$ in $P=(1,1,2)$ using implicit differentiation. My idea is to calculate $\frac{∂z}{∂x}$ in $P(1,1,2)$ and $\frac{∂z}{∂y}...
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Implicit differentiation of an equation of a hyperbola

Prove that an equation of the tangent line to the graph of the hyperbola : $(x^2/a^2) - (y^2/b^2) = 1$ at the point ($x_0$, $y_0$) is $x x_0/a^2 - y y_0/b^2 = 1$ (1) I implicitly ...
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1answer
35 views

Implicit Differentiation - Different Approaches

Given is $F(x,y)=ye^{3x}-2x^2=0$ I was asked to calculate $y’$ using implicit differentiation. I know that $y’=-\frac{Fx}{Fy}=-\frac{\frac{∂F}{∂x}}{\frac{∂F}{∂y}}.$ So I obtained: $y’(x)=-\frac{3ye^...
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How to differentiate, for example, f(r,t ) wrt r without knowing in what way f depends on r and t?

Say we have some function $\psi'(r,t)$, given by $\psi'(r,t)=e^{af(r,t)}\psi(r,t)$ and we want to calculate $\nabla^2\psi'(r,t)$. I obviously know how to do all the basic steps of this product rule ,...
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$y=f(x) \in C^1$ is defined implicitly by $ax + by = f(x^2+y^2), f'(x) = ?$

Problem $y=f(x) \in C^1$ is defined implicitly by $ax + by = f(x^2+y^2)$. $a$ and $b$ are constants . $f'(x) = ?$ Analysis The answer for this exercise given by my teacher is $$f'(x) = \frac{2xf'(x^...
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1answer
32 views

Calculus - Implicit Curve - rearranging derivative equation

I'm new to this community, so please bear with me if I make a mistake following rules and conventions, and thanks in advance. I'm currently learning - following along from this video ("Implicit ...
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2answers
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Differentiating with respect to size of index

I have the following function: $$a\sum_{i=1}^{n}x_i. $$ I wish to differentiate with respect to $n$. If all $x_i$s were the same, this problem would be trivial, obviously. Can anyone help?
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Having trouble solving implicit function

$$ x\cos y + y\cos u + u\cos x = 1 $$ I'm trying to calculate u$'_x$ and $u'_y$ in the point $(0,1)$ while $0 \leq u \leq \pi$. I've managed to find: $$u'_x = - \frac{\cos y-u\sin x}{\cos x-y\sin u}...
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1answer
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Calculating the second order partial derivatives of $z(x,y.)$. [closed]

If we know that $z_{x}^{'} = \frac{yz}{z^{2} - xy },$, how can I calculate the second order partial derivative with respect to x, knowing that the final answer should be (as given at the back of the ...
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1answer
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Calculating the first order partial derivatives of $z(x,y.)$.

My function is $z^3 - 3xyz = 1$ and I calculated $z_{x}^{'}$ and I got $z_{x}^{'} = \frac{yz + yy^{'}z}{z^{2} - xy }$. but the answer at the back of the book is $z_{x}^{'} = \frac{yz}{z^{2} - xy },$ ...
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Implicit Differentiation - Logarithm

$x\log(x) + y\log(y) = 1$ $\dfrac{dy}{dx}= ?$ I calculated $\frac{dy}{dx}= -\frac{1+\log(x)}{1+\log(y)}$ however, the correct answer seems to be $-\log(x)/\log(y)$ I'm confused, can someone ...
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Is this the correct way to do Implicit Differentiation?

Problem: Use implicit differentiation to compute $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ of the function $ x^3 + y^3 +z^3 - 3xyz = 0 $. What I Got: $3x^2 + 0 ...
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1answer
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Quotient rule and implicit differentiation

Find $\frac{dy}{dx}$ for $x^2=\frac{x-y}{x+y}$. I have solved this in two ways. First, I multiplicated the whole equation by $x+y$ and then I calculated the implicit derivative. I got the ...
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Implicit differentiation $\sin(xy)$

When I check my answer using the implicit differentiation tool on wolframalpha.com, I get a result I can't agree with. So I'd like to hear your opinion :) Asked: use implicit differentiation to ...
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Folium of Descartes derivative at $0$

With regard to this curve: $$3xy=x^3+y^3$$ I understand that $\frac{dy}{dx}$ is not defined at $(0,0)$, but, there must be some more information right as there are $2$ tangent lines. I know my ...
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36 views

Assumption in deriving implicit function theorem?

I'm have a bit of trouble understanding how dependence of variables work with implicit functions. This troubles me during the chain rule and specifically in this case, the implicit function theorem. ...
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3answers
40 views

Trig Differentaition

Differentiate $y=27 \sec^3(x)$ with respect to $x$. I tried splitting the $\sec^3(x)$ into $\sec^2(x)\cdot \sec(x)$ and using the product rule but that didn't work.
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Differentiate $e^{y/x} = 20x-y$

I am trying to use implicit differentiation to differentiate $e^{y/x} = 20x-y$. I get $\frac{20}{2 e^{y/x} \cdot \frac{x-y}{x^2}}$, but according to the math website I'm using, "WebWork", this is ...
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Implicit differentiation in multivariable calculus

What I don't understand is the disconnect between the reasoning for implcit derivation in single variable and multivaraible calc. in single variable calc they define say $y = f(x)$ for a small ...
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5answers
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Differentiate $11x^5 + x^4y + xy^5=18$

I am not sure how to differentiate $11x^5 + x^4y + xy^5=18$. I have a little bit of experience with implicit differentiation, but I'm not sure how to handle terms where both variables are multiplied ...
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2answers
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How to convert a rate involving radians to something that can be applied to a straight direction in a related rates problem.

I can do related rates problems a little bit, but I've been given one that requires me to use a rate of $\frac{-\pi}{6}$ radians per second to figure out how fast a plane is going. Since I assume that ...
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Total Differential of an Equation

I want to find the total differential of an equation which has been defined as: $ Y = C((1-t)Y, M/P)$ where $t$ is a parameter and $M$,$P$, and $Y$ are variables. And Y is a function of C which in ...
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1answer
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Does Implicit Differentiation Depend on the Form of the Equation?

I stumbled across a related rates problem which involved using implicit differentiation: A rock is initially dropped at height h a horizontal distance d from a street lamp that's H tall. The lamp ...
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29 views

$f(x+\frac{y}{2})-f(x-\frac{y}{2})=2x^2y+5y^2$. Find $\frac{d f(3)}{dx}= f'(3)=?$

$f(x+\frac{y}{2})-f(x-\frac{y}{2})=2x^2y+5y^2$ $\frac{d f(3)}{dx}= f'(3)=?$ As there is no information on whether $y$ is a function or a constant, I believe it must be treated as a constant. ...
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28 views

Implicit Differentation of Polar Coordinates

"Let D be $R^2$ \ {$(x,0) | x \leq 0$}. The polar angle gives for each $(x,y)$ a value $\theta(x,y)$ in the interval $(-\pi, \pi)$. The function $\theta$ is continuous and differentiable on D. a) ...
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1answer
30 views

Partial derivative implicit differentiation

dz/dx = ? cos(zy)+zx^2 = (1+y)e^(x-z) For the left side through implicit differentiation I have found (-sin(zy))(y*(dz/dx))+2xz+(dz/dx)x^2. I am completely unsure how to approach the right side, ...
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Solving for points such that the tangent is parallel to the x-axis on a lemniscate.

I am asked to find ${y}'$ of $(x^{2}+y^{2})^{2} = a^{2}(x^{2}-y^{2})$ with $y(x)$ and $a$ as a positive constant, which is given in the solutions as: $ {y}'(x) = \frac{(a^{2}-2(x^{2}+y^{2}))x}{(2(x^...
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1answer
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Can implicit derivatives exist at points where an equation is not satisfied? [closed]

For example, given the equation $x + y - z + \cos(xyz) = 0$. Is it possible to find partials of $z$ w.r.t. $x$ and $y$ at the point $(0,0,0)$?
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57 views

Find the slope of the tangent line to the curve defined by $7x^4 - 8xy - 6y^3 = 322$ at the point $(2, -3)$

Find the slope of the tangent line to the curve defined by $$ 7x^4 - 8xy - 6y^3 = 322$$ at the pont $(2, -3)$ I'm having a tough time using implicit differentiation and chain rule with all ...
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0answers
15 views

Implicit differentiation to obtain expression value

Question and attempt at the question I've been trying to evaluate the following expression, I'm not sure if I'm heading into the right direction though. Could someone kindly let me know what the ...
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0answers
48 views

Finding implicit derivative $\frac{∂z}{∂x}$ in multivariable equation

I am pretty new to multivariable calculus, I know how to find $f_x$, $f_{xx}$, $f_{xy}$ etc. but that's about all. I want to solve this question: Find the value of $\frac{∂z}{∂x}$ at the point (1, ...
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25 views

Given that $f(y/x,z/x) = 0$ defines $z$ implicitly as $z=g(x,y)$, show $x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = g(x,y).$

Given that the equation $f(y/x,z/x) = 0$ defines $z$ implicitly as the function $z=g(x,y)$, show that $$x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = g(x,y)$$ at points where $...
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0answers
29 views

Find the coordinates of the stationary points of $e^x +ye^{-x} = 2e^2$.

Find the coordinates of the stationary points of $e^x +ye^{-x} = 2e^2$. So I have differentiated this implicitly, which I think is correct but I'm then unsure how I'd actually solve the equation. $$\...