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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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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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28 views

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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2answers
75 views

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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19 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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39 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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If $5 x^2 + y^4 = 21$ then evaluate the second derivative of y with respect to x when x = 2 and y = 1. [closed]

If $5 x^2 + y^4 = 21$ then evaluate the second derivative of y with respect to $x$ when $x = 2$ and $y = 1$. Round to two decimal places.
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75 views

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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21 views

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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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
34 views

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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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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1answer
27 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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27 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
29 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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1answer
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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
10 views

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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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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14 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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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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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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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. $$\...
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1answer
31 views

Implicit function question

Explain why $\sqrt{x^2-y^2}+\arcsin(x/y)=0$ does not define $y$ as an implicit function of $x$. Quite confused by this, mainly because I do not fully understand really what it means for an equation ...
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1answer
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Deriving $dy/dx = 2\cos x/\cos y$ given $\sin y=2\sin x$

My original question is to find the second derivative of $\sin y=2\sin x$ I derived it once got $2\cos x/\cos y$ which was correct but the second time did not get $3\sec^2y\tan y$ which is the ...
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1answer
56 views

Challenging differentiation problem

I am trying to do it with substitution and getting a $1$ as answer but the answer is $\ln (4e^2)$ Given: $$ y = (1 + \frac{1}{x})^x + x^{1+\frac{1}{x}}$$ Evaluate $y'(1)...
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3answers
41 views

For the equation given, evaluate y′ at the point (−2,−1) [closed]

I must find the derivative for this equation $$ (3x−y)^4+4y^3=621 $$ at the point $(-2,-1)$.
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finding partial derivatives from equations

Picture of the problem Quantities u,v,w and z satisfy the equations $2*z+2*w+v+3*u=-3$, $z+3*w+2*v+5*u=6$ Find the partial derivatives $(\frac{∂z}{∂w})_z$ and $(\frac{∂v}{∂z})_w$ at $(0,3)$ (...
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How or why are implicit functions actually functions?

I'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function. I didn't quite get it at that time so I decided to ...
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1answer
22 views

Differentiating a function that is defined generally (for eg, $ f (x_1, x_2, … x_n) $)

If I'm asked to differentiate a function that is simply of the form: $ f(tx_1, tx_2, ... tx_n) $ (i.e., the function is not "defined" to be particularly anything), how should I go about this? I want ...
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2answers
100 views

How to find the derivative of $cos 5x$ using first principle

So my textbook solution uses some discrete method to arrive at -5sinx 5x. But I want to know a simpler technique to get the solution. Can anyone post?
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1answer
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implicit differentiation yielding different expressions

I was studying Thomas's Calculus book and attempted a question using implicit differentiation. $$x^3=\frac{2x-y}{x+3y}$$ I differentiated both sides directly, using the quotient rule on the RHS to ...
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Is the differential forms perspective on $dx$ incompatible with the technique of implicit differentiation?

Suppose $$x^2 + y^2 = 5^2.$$ We're trying to find $dy/dx$ at $(3,4).$ Applying $d$ to both sides: $$2x dx + 2y dy = 0$$ Or in other words: $$2x dx + 2y dy = 0dx + 0dy$$ Since the covectors $dx_p$ ...
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Canonical norm on cartesian product of vector spaces

Let $(V_i)_{i=1,...,n}$ be a finite ordered set of normed vector spaces over $\mathbb R$. After searching, there doesn't seem to be a consensual choice of a standard norm on the product space $V= V_1 \...
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I'm differentiating this wrong!

I am self-teaching calculus and have been looking at the related rates practice problems here: https://www.whitman.edu/mathematics/calculus_online/section06.02.html I am having trouble with the last ...
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how to tell the monotonicity of an implicit function?

I have a function $F(x,y)=0$. Taking derivative of this implicit function, i can get that $y'>0$ if $x>Cy$, and $y'\le 0$ if $x\le Cy$. $C$ is some constant. Can i say $y$ is concave in $x$?
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Why is dx/dt = -(∂u/∂t) / (∂u/∂x)? [closed]

I found that $\frac{dx}{dt} = -\cfrac{ \frac{\partial u}{\partial t} }{ \frac{\partial u}{\partial x}}$ on the internet. I can´t figure out if it is true and why.
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Rules for inverse functions and partial derivatives

I have that $u(t,x)$ satisfies $\partial u/\partial t + u \cdot \partial u/\partial x = 0$ I need to show that if $x = x(t)$, then $dx/dt = u(t,x)$ So far I have $u = -\partial u/\partial t \...
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2answers
43 views

Implicit Differentiation with a Tangent Line

I was looking to implicitly differentiate $$-22x^6+4x^{33}y+y^7=-17$$ and found it to be $$\dfrac{dy}{dx}=\dfrac{132x^5-132x^{32}y}{4x^{33}+7y^6}$$Now, I am trying to find the equation of the tangent ...
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3answers
29 views

Differentiation for a function in the integral form.

I want to know generally how we differentiate a function $F(x)$ in the following form, $$F(x)=\int_a^x f(x,t)dt$$ For example, if we can work out the explicit form of $F(x)$ as the example below $$F(x)...
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3answers
120 views

Confusion about implicit differentiation $\frac{dy}{dx}$

Today I learnt about implicit differentiation using this: $\frac{d}{dx}(f)$ = $\frac{df}{dy} \times \frac{dy}{dx}$ I don't understand when doing implicit differentiation how the d/dy part works for ...
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2answers
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Find coordinate in first quadrant which tangent line to $x^3-xy+y^3=0$ has slope 0

Find coordinate in first quadrant which tangent line to $x^3-xy+y^3=0$ has slope 0 First, I do implicit differentiation: $\frac{3x^2-y}{x-3y^2}=y'$ so I look at the numerator and go hmmm if i put ...
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1answer
53 views

Partial derivative calculation for circular or iterative functions

I have a question of partial derivatives for implicit functions. I have three equations: $\rho_k=\rho_l\frac{\lambda_l}{H_l}+\rho_g\frac{(1-\lambda_l)}{1-H_l}$ $Re=\frac{\rho_kv_m\phi}{\mu}$ $H_l=f(...
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Hai… my problem is i dont know how to solve this question because I confuse to use any method

$e^{n+y}=1+5\sqrt{y^2}-\ln\left[\frac{y^5(6n^2+1)}{\sqrt{2n^3-4}}\right]$ Find $\frac{dy}{dn}$ when $n=0$ and $y=1$. How can I approach this problem? (original link: https://i.stack.imgur.com/W9Mym....
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Total derivative of implicit functions

If we have three implicit functions as following: $$f(x,y,z)=0$$ $$g(x,t)=0$$ $$M(x,y)=0$$ First, if I want to compute the derivative of z w.r.t x , will it be as following? $$\frac{dz}{dx}=z_x+z_y\...
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266 views

Optimization Problem. Find Smallest Perimeter Given Area.

QUESTION Find the dimensions of a rectangle with area $1000$m$^2$ whose perimeter is as small as possible. MY WORK I think we are solving for $\frac{dy}{dx}$: \begin{align*} P &= (2x+2y) \\ ...
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1answer
26 views

Basic Implicit Differentiation Problem

QUESTION: Find $\frac{dy}{dx}$ by implicit differentiation. $x^2-4xy+y^2=4$ MY SOLUTION $\frac{d}{dx}x^2 - \frac{d}{dx}4xy + \frac{d}{dx}y^2 = \frac{d}{dx}4$ $2x-(4x)'(y)+(4x)(y)'+2y\frac{dy}{dx} =...
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2answers
49 views

proof of log derivative using implicit differentiation [closed]

How would you use implicit differentiation and the fact that $\log_b x$ is the inverse of $b^x$ to prove that $$\frac {d}{dx} (\log_b x) =\frac{1}{(\ln b)x}$$
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2answers
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Implicit differentiation at a point with trigonometry and a fraction.

Find $\frac{dy}{dx}$ for $xcosy-2sin(\frac{y}{2})=0$ at (2,$\frac{\pi}{3}$) I tried using the power rule and chain rule but I do not seem to solve the problem. Can someone tell me how to solve it?.
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2answers
34 views

How to Complete Implicit Differentiation Problem

The problem I am trying to find $\frac{dy}{dx}$ of: $$\sqrt{x+y}= x^4 + y^4$$ I have attempted to solve the problem via the following steps: $x^{1/2} + y^{1/2} = x^4 + y ^4$ $\frac{d}{dx}x^{1/2}+\...
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Related Rates / Derivatives [closed]

Two aircrafts are in the same airspace, with Plane A 500 km south of Plane B. If Plane A is travelling 600 km/h due south while Plane B is travelling 800 km/h due west, determine how quickly the ...