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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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Implicit differentiation choice

I was reading Calculus early transcendentals by Howard Anton, in which I encountered an example as follows, Find the slope of tangents of a sphere $x^2+y^2+z^2=1$ in the direction of $y$ at points $(2/...
Kaustubh Limaye's user avatar
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How to find the gradient when it equals to "$\frac00$"?

Find the gradient of the curve $C:ay^3+bx^2y+cxy=1$ where $$cy+2bxy=0,\quad cx+bx^2+3ay^2=0,$$ if such a point exists. If we try implicit differentiation, we get $$3ay^2y'+2bxy+bx^2y'+cy+cxy'=0,$$ ...
youthdoo's user avatar
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Second order partial implicit derivatives

I have some function $F(x, y, z) = 0$, and wish to find the second order cross derivative $\frac{d^2z}{dxdy}$. I've easily been able to get the second order derivatives $\frac{d^2z}{dx^2}$ and $\frac{...
sprw121's user avatar
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Geometric interpretation of implicit differentiation

It is well known that, given a function $f:\mathbb{R} \to \mathbb{R} $, $f'(x_0)$ can be interpreted as the slope of the tangent line to $f$ in $x_0$. What about curves of the form $F(x, y, c)=0$, ...
Davide Masi's user avatar
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If implicit differentiation yields an undefined expression at a point. Does this mean that derivative is undefined at that point?

Equation $(x-y)^2=0$ implicitly defines two functions: $y=x$. The derivative of each is 1. But the implicit differentiation yields the expression that not defined at $y=x$: $$(x-y)^2=0$$ $$\frac{d}{dx}...
Nikolay Isaev's user avatar
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40 views

Simple Optimization Problem from Economics

An individual has 100 dollars that they have to invest in Asset 1 or 2. The returns from the two assets are $r1\stackrel{d}{=} c + v_1, r2=v2$ for some constant $c$ and independent random variables $...
Juanito's user avatar
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Differentiate $x=rcos(\theta)$ with respect to y

So we know: $x=rcos(\theta)$, $y=rsin(\theta)$ and $x^2 + y^2 = r^2$. I assume it will help to consider $r$ and $\theta$ as functions of $x$ and $y$, but I am not sure how to incorporate this.
PeakyBlaze7788's user avatar
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17 views

Apply the implicit function theorem of the ratio $c_{t+1}/c_t$ that itselfs depend on the level $c_t$

I'm trying to compute the derivative $\frac{d c_{t+1}/c_t}{d (1+r_{t+1})}$ of this function: $$\frac{c_{t+1}}{c_{t}} = \left( \beta (1+r_{t+1}) +\gamma \frac{(s-c_{t}+z) ^{-\Sigma }}{c_{t+1}^{-\sigma}}...
Mr. Fafa's user avatar
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Moving expressions around gives different implicit differentiation

Here's an exercise from Thomas Calculus, I need to do an implicit differentiation: $$ x^3=\frac{2x-y}{x+3y} $$ If I enter this in Wolfram Alpha, I get: $$ y'(x) =-\frac{3 x^4 + 18 x^3 y + 27 x^2 y^2 - ...
Stanislav Bashkyrtsev's user avatar
3 votes
1 answer
88 views

Total derivative of f(x, g(x, y)) and its approximation

I understand the steps to calculate the total derivative of f(x, g(x)) Related: Derivative of $f(x, g(x))$ with respect to $x$ I have three sub-questions related to calculating the total derivative of ...
Julie Taylor's user avatar
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1 answer
50 views

Finding the second derivative of $1.51x^2 + y^2 = 1 + 0.71x^2y^2$ to calculate the curvature

Consider the equation $$1.51x^2 +y^2 = 1 + 0.71x^2y^2.$$ In this question you will calculate the curvature, $\rho$. Evaluate the derivative at the point described — you should get decimal numbers $$x= ...
Lollipop 's user avatar
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Assume that $\frac{d}{d\theta}\sin\theta = \cos\theta$. Use implicit differentiation to prove that $\frac{d}{d\theta}\cos\theta = - \sin\theta$

Assume that $\frac{d}{d\theta}\sin\theta = \cos\theta$. Use implicit differentiation to prove that $\frac{d}{d\theta}\cos\theta = - \sin\theta$ Here's my attempt: $(\sin\theta)^2 + (\cos\theta)^2 = 1$...
ten_to_tenth's user avatar
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Optimization word problem involving perimeter and area of an arched window

I'm working through an optimization problem in my textbook about maximizing the area of a rectangular window with an arched top. My question is concerns how to think about a certian variable. I'm torn ...
El Jfe's user avatar
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Help understanding implicit derivation

I'm currently enrolled in Calculus 1, and everything has been pretty smooth up until these last two sections involving the chain rule and implicit derivation. After watching multiple YouTube videos, ...
gabrielz's user avatar
1 vote
1 answer
61 views

Can you square root both sides of an implicit equation to get the derivative? [closed]

Suppose you have $x^2=(4x^2y^3 + 1)^2$, can you square root both sides of the equation to make it simpler?
mathingishard's user avatar
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1 answer
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If $f(x)=\frac{2}{\sqrt{3}}\int_{0}^{\sqrt{3}}f\left(\frac{\lambda^2x}{3}\right)d\lambda,x>0$ and $f(1)=\sqrt3$ and if $f(\alpha)=6$ . find $\alpha$

Let $f(x)$ be a differentiable function satisfying $$f(x)=\frac{2}{\sqrt{3}}\int_{0}^{\sqrt{3}}f\left(\frac{\lambda^2x}{3}\right)d\lambda,x>0$$ and $f(1)=\sqrt3$ . If $y=f(x)$ passes through the ...
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Partial derivative of endogenous variables and total differentiation

I am given two equations: $x=\frac{u}{v}(1- \frac{1-y+py}{p})$ and $y\frac{x}{2}= a\frac{p}{1-y+py}$ where $p$, $a$, $u$ and $v$ are parameters. I am asked to provide $\frac{\delta x}{\delta y}$ and $\...
learner's user avatar
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4 votes
1 answer
125 views

Implicit differentiation gives different answer

I'm trying to implicitly differentiate the following equation: $\frac{x+f(x)}{c+f(x)}=4$ where $c$ is a constant. Here's what I get: $$ \frac{\mathrm d}{dx}\left(\frac{x+f(x)}{c+f(x)}\right)=\frac{\...
Jasper1378's user avatar
3 votes
2 answers
286 views

Strange results from implicit differentiation

Disclaimer: this is probably just a giant misunderstanding of implicit differentiation, but I'm not sure where! I'm looking at an initial-value problem for a simple damped harmonic oscillator but ...
ManRow's user avatar
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How do I solve: The normal to the curve $\ xe^{-y} + e^{y} = 1 + x$ at the point $(c, \ln c)$ has a $y$-intercept $\ c^2+ 1$. Determine c.

How do I solve: The normal to the curve $\ xe^{-y} + e^{y} = 1 + x$ at the point $(c, \ln c)$ has a $y$-intercept $\ c^2+ 1$. Determine c. I tried substituting $\ x=0$ to find out the y-intercept, but ...
user1271556's user avatar
-1 votes
2 answers
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Prove the orthogonality of these curves [closed]

I am having trouble proving the orthogonality of the following curves: $$\begin{cases} x = \frac{1}{2}(v_1^2 - v_2^2), v_1=\text{const} \\ y = v_1v_2, v_2=\text{const} \end{cases}$$ Here's everything ...
Bagaringa's user avatar
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2 answers
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Multi-variate cross-partial derivative of a inverse function

I have an invertible mapping $y=f(x,\theta)$ where $x,y\in\mathbb{R}^K$ with a scalar parameter $\theta\in\mathbb R$. Consider its inverse $x=g(y,\theta)$. I'm interested in the matrix $\partial^2g/\...
Kirill Borusyak's user avatar
1 vote
0 answers
38 views

How is rate of change dx/dt in ladder problem doesn't match the actual rate of change.

The pictures above describes the question. We have to find the rate of change in x-axis direction. The answer is derived from implicit differentiation and is $4/3$. The process is: [y(t) gives y-axis ...
user avatar
1 vote
1 answer
90 views

Implicit differentiation of $\ln(x^2+xy+y^2)=1$ [closed]

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ of $\ln(x^2+xy+y^2)=1$. So for this natural log should I start by using the power rule or the the product rule of $xy$? The textbook talks about taking the ...
Leila Khodor's user avatar
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27 views

Is the Primer Introduction to Derivation of Implicit Functions Using This Definition Ambiguous?

I'm exploring the concept of implicit function derivation and parameterized curve derivation, and I've encountered some points that seem vague to me. I would appreciate any insights or clarifications ...
g_prole's user avatar
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2 votes
0 answers
76 views

Find the derivative of $x^{\sqrt{y}}=y^{\sqrt{x}}$.

I am trying to solve: $x^{\sqrt{y}}=y^{\sqrt{x}}$ Here's my solution. Please correct me if there is an error and kindly explain why. $\sqrt y\ln{x}=\sqrt x\ln{y}$ $\left[y^\frac{1}{2}\ln{x}\right]^\...
PRD's user avatar
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5 votes
3 answers
137 views

What exactly does $\frac{\mathrm{d}y}{\mathrm{d}x}$ calculate in implicit differentiation? (details given)

Does it calculate the derivative of the given equation? Because in implicit differentiation we don't have $y= equation$ so I don't know what we are differentiating using the $\frac{\mathrm{d}y}{\...
ATS's user avatar
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0 answers
51 views

Implicit Differentiation: Proving a point is perpendicular at a certain point on two interesecting tangents.

Given the following question: Consider the curves C1 and C2 defined as follows; C1 : xy = 4 , x > 0 C2 : y^2 - x^2 = 2 , x > 0 Let P(a,b) be a unique point where the curves C1 and C2 intersect. ...
Liam Gannaway's user avatar
2 votes
2 answers
65 views

How do I implicitly differentiate $\frac{d}{dx}(2x(y')^2y'')$

I know that I will probably need to use the 3-way chain rule: $$(fgh)'=f'gh+fg'h+fgh',$$ but since I'm differentiating in terms of $x$, I'm very confused as to how to to "append" $y'$ terms ...
Holland Davis's user avatar
1 vote
1 answer
59 views

Second implicit derivative

Given $$x^2y+3y^3=7$$ I have to evaluate $y''$ at $(x,y)=(2,1)$. I know $$y'=-\frac{2xy}{x^2+9y^2}$$ hence, at $(x,y)=(2,1)$, $$y'=-\frac{4}{13}$$ Therefore, $$y''=-\left(\frac{(2xy)'(x^2+9y^2)-(2xy)(...
mvfs314's user avatar
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0 votes
1 answer
118 views

Understanding Implicit Differentiation (intuitive)

Let's say we have: $$ x^2 +y^2 = 25 $$ definition of differentiation: $$\lim_{x\to h} = \frac{f(x+h)-f(x)}{h}=\frac{dy}{dx}$$ when we do implicit differentiation we have: $$ df(x^2 +y^2) =df(25) $$? ...
VOZ ESTOICA's user avatar
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2 answers
110 views

How do derivatives even work?

I'm a student currently looking into partial derivatives and was writing notes for a simple example. $$\require{cancel}\text{find the derivative with respect to x:}\\x^2+xy-y^2=8$$ I got to the point ...
Nathan Williams's user avatar
1 vote
1 answer
421 views

Difference between the derivatives of arcsin(x) and arccos(x)

The derivative of $\arccos(x)$ and $\arcsin(x)$ are respectively $-\frac 1{\sqrt{1-x^2}}$ and $\frac 1{\sqrt{1-x^2}}$. However in the classic proof using implicit differentiation, when taking the ...
DrVendetta's user avatar
0 votes
1 answer
61 views

Need help with Implicit differentiation for partial derivative

I am currently taking Cal 3 and my professor taught in class that if I have a function $F(x,y,z)=c$ (where $c$ is a constant), then $\frac{dz}{dx}=-\frac{F_x}{F_z}$. I understand the textbook ...
Minh Tri Pham's user avatar
3 votes
1 answer
188 views

Calculating the functional derivative of an implicit functional

We want to calculate the functional derivative of the following wrt $\rho$: $$F=\int X dx$$ where $X$ is implicitly defined as: $$X=\frac{1}{1+\overline \rho(x) \int \rho(x) X dx}$$ and $\overline \...
Ali's user avatar
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0 votes
1 answer
90 views

implicit differential equation, y is a solution [closed]

Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution....
Amy Wang's user avatar
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0 answers
29 views

Comparative Statics, implicit functions and second order derivatives.

I have that $x^{*}(w,z)$ and $y^{*}(w,z)$ is the implicit solution to a the system $F(x^{*}, y^{*},w) = 0$ and $H(x^{*}, y^{*},z) = 0$. Using the implicit function theorem, I can compute $\frac{\...
Karl Smith's user avatar
2 votes
1 answer
120 views

Why do we call $y$ a function of $x$ in implicit differentiation?

When we have something like $y = 2x$ we understand $y$ to be the value of the function $f$ at each point $x$ where $f(x) = 2x$, to reiterate, $y$ is not a function but merely a label for the output of ...
Nav Bhatthal's user avatar
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0 votes
0 answers
60 views

Mapping the derivative of an implicit function on a 2D plane

I'm in the middle of my first calculus class, a week ago we covered how to find the derivative of implicit functions, and I'm still thinking about it. I completely understand what I am supposed to ...
accoustician's user avatar
1 vote
2 answers
61 views

Stuck on implicit differentiation exercise; inquiring about how to approach it

On exercise 8 on section 3.6 of Stewart’s Second Edition Single-Variable Calculus, it is asking me to find the derivative of y with respect to x by implicit differentiation. The problem it gives me is ...
Aaron Garcia's user avatar
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0 answers
36 views

Multivariable implicit differentiation, $U=T\frac{\partial{S}}{\partial{T}}dT$

Given the equation $U=T\frac{\partial{S}}{\partial{T}}dT$ where $S$ is a function of V and T and V is being held constant, how do we then find $\frac{\partial U}{\partial T}$? The way I'm explaining ...
James H's user avatar
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-1 votes
1 answer
22 views

Is there an order to approaching implicit differentiation questions?

For example, I have the question "solve for $\frac {dy}{dx}$: $$ycos(xy)=y^2+2"$$ and I'm not sure if I should differentiate with the product rule or the chain rule first for the left side of the ...
ilya.te's user avatar
  • 13
0 votes
2 answers
74 views

Use implicit differentiation to find dy/dx for $x \ge -2$. $y\sqrt{x+2}=xy+3$; I am getting a different answer than is given

$$y\sqrt{x+2}=xy+3$$ find $\frac {dy}{dx}$ for $x \ge -2$. The given answer is $$-\frac{y(2(x+2)^{1/2}-1)}{2(x(x+2)^{1/2}-x-2)}$$ Here's my work so far: $$y\sqrt{x+2}=xy+3$$ $$y\frac d{dx}\sqrt{x+2}+\...
ilya.te's user avatar
  • 13
2 votes
3 answers
405 views

What happens when we take the derivative of an implicitly defined function?

Consider the function y(x) implicitly decided by this equation: $\sin(xy)=\cos(x+y)$. The graph on plane looks like this: According to my textbook, we can take the derivative of both sides with ...
user1206899's user avatar
3 votes
4 answers
418 views

How do I find the horizontal tangents to the curve $x^3 + y^3 = 6xy$?

How do I find the horizontal tangents to the curve $x^3 + y^3 = 6xy$? James Stewart in section $3.6$ of the 7th edition (on page $167$) shows in a straightforward way that there's a horizontal tangent ...
user1145880's user avatar
2 votes
1 answer
231 views

Chain rule (multivariate) and implicit differentiation problem

I have been worked out a solution for the following problem, but I am wondering if there is an easier way to solve it. I would be very grateful for any suggestions! Let $z=f(x,y)$ a function of two ...
Apollo13's user avatar
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1 vote
1 answer
44 views

If $y=f(x)$ such that $(x+1)^3+(y+1)^3=16$ then evaluate : $\frac{d}{dx}\left(f(x+g(x)).g(x+f(x))\right)$ at $x=-1+\sqrt[3]{15}$

Let $f:\mathbb{R}\to\mathbb{R};y=f(x)$ be an explicit function defined by the implicit equation $$(x+1)^3+(y+1)^3=16$$ Let $g$ be the inverse of $f$. Evaluate : $$\frac{d}{dx}\left(f(x+g(x)).g(x+f(x))\...
Maverick's user avatar
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3 votes
1 answer
296 views

Finding the area surrounded by a part of the implicit equation $\sin (y^x) = \cos (x^y)$

Finding the area surrounded by the part of the implicit equation $\sin (y^x) = \cos (x^y)$ such that $y\le 2n-x$ where $n$ is the solution to $n^n=\frac{\pi}{4}$ where $n<0.5$ bounded by the $x$ ...
Dylan Levine's user avatar
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1 vote
0 answers
32 views

Computation of smooth Cholesky factorizations

Assume $X(t)$ is a time dependent positive definite symmetric matrix which satisfies the matrix differential equation $\dot{X}(t) = Q(X(t)), X(0) = 0, $ where $Q(X(t))$ is a symmetric positive ...
A Varga's user avatar
  • 11
1 vote
1 answer
163 views

Solving logarithmic derivative in implicit form

I have the following function $$ a\text{log}\left(X\right)+b\text{log}\left(y\right)=c\left[d\text{log}\left(X\right)+e\text{log}\left(y\right)\right]+\text{log}\left(\frac{A}{B}\right) $$ I wish to ...
Kwame Brown's user avatar

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