Questions tagged [implicit-differentiation]
For questions on finding and evaluating derivatives when a function is defined implicitly.
1
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1answer
30 views
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 ...
1
vote
3answers
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 ...
2
votes
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 ...
0
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0answers
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.
...
2
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3answers
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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0answers
23 views
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.
3
votes
3answers
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 ...
0
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0answers
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 ...
4
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5answers
76 views
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 ...
2
votes
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 ...
0
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0answers
14 views
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 ...
0
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1answer
18 views
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 ...
0
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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. ...
0
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0answers
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) ...
0
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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, ...
0
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1answer
20 views
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^...
0
votes
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 $(0,0,0)$?
0
votes
2answers
46 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 ...
0
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0answers
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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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, ...
0
votes
0answers
22 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
24 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.
$$\...
2
votes
1answer
31 views
Implicit function question
Explain why $\sqrt{x^2-y^2}+\arcsin(x/y)=0$ does not define $y$ as an implicit function of $x$.
Quite confused by this, mainly because I do not fully understand really what it means for an equation ...
1
vote
1answer
34 views
Deriving $dy/dx = 2\cos x/\cos y$ given $\sin y=2\sin x$
My original question is to find the second derivative of $\sin y=2\sin x$
I derived it once got $2\cos x/\cos y$ which was correct but the second time did not get $3\sec^2y\tan y$ which is the ...
-1
votes
1answer
56 views
Challenging differentiation problem
I am trying to do it with substitution and getting a $1$ as answer but the answer is $\ln (4e^2)$
Given:
$$ y = (1 + \frac{1}{x})^x + x^{1+\frac{1}{x}}$$
Evaluate $y'(1)...
0
votes
3answers
41 views
For the equation given, evaluate y′ at the point (−2,−1) [closed]
I must find the derivative for this equation
$$
(3x−y)^4+4y^3=621
$$
at the point $(-2,-1)$.
0
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0answers
34 views
finding partial derivatives from equations
Picture of the problem
Quantities u,v,w and z satisfy the equations
$2*z+2*w+v+3*u=-3$,
$z+3*w+2*v+5*u=6$
Find the partial derivatives $(\frac{∂z}{∂w})_z$ and $(\frac{∂v}{∂z})_w$ at $(0,3)$
(...
2
votes
3answers
38 views
How or why are implicit functions actually functions?
I'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function.
I didn't quite get it at that time so I decided to ...
0
votes
1answer
22 views
Differentiating a function that is defined generally (for eg, $ f (x_1, x_2, … x_n) $)
If I'm asked to differentiate a function that is simply of the form: $ f(tx_1, tx_2, ... tx_n) $ (i.e., the function is not "defined" to be particularly anything), how should I go about this? I want ...
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votes
2answers
100 views
How to find the derivative of $cos 5x$ using first principle
So my textbook solution uses some discrete method to arrive at -5sinx 5x. But I want to know a simpler technique to get the solution. Can anyone post?
1
vote
1answer
27 views
implicit differentiation yielding different expressions
I was studying Thomas's Calculus book and attempted a question using implicit differentiation.
$$x^3=\frac{2x-y}{x+3y}$$
I differentiated both sides directly, using the quotient rule on the RHS to ...
9
votes
1answer
116 views
Is the differential forms perspective on $dx$ incompatible with the technique of implicit differentiation?
Suppose $$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$$
Since the covectors $dx_p$ ...
0
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0answers
55 views
Canonical norm on cartesian product of vector spaces
Let $(V_i)_{i=1,...,n}$ be a finite ordered set of normed vector spaces over $\mathbb R$. After searching, there doesn't seem to be a consensual choice of a standard norm on the product space $V= V_1 \...
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0answers
75 views
I'm differentiating this wrong!
I am self-teaching calculus and have been looking at the related rates practice problems here: https://www.whitman.edu/mathematics/calculus_online/section06.02.html
I am having trouble with the last ...
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0answers
24 views
how to tell the monotonicity of an implicit function?
I have a function $F(x,y)=0$. Taking derivative of this implicit function, i can get that $y'>0$ if $x>Cy$, and $y'\le 0$ if $x\le Cy$. $C$ is some constant. Can i say $y$ is concave in $x$?
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2answers
56 views
Why is dx/dt = -(∂u/∂t) / (∂u/∂x)? [closed]
I found that $\frac{dx}{dt} = -\cfrac{ \frac{\partial u}{\partial t} }{ \frac{\partial u}{\partial x}}$ on the internet. I can´t figure out if it is true and why.
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0answers
24 views
Rules for inverse functions and partial derivatives
I have that $u(t,x)$ satisfies
$\partial u/\partial t + u \cdot \partial u/\partial x = 0$
I need to show that if $x = x(t)$, then $dx/dt = u(t,x)$
So far I have
$u = -\partial u/\partial t \...
2
votes
2answers
43 views
Implicit Differentiation with a Tangent Line
I was looking to implicitly differentiate $$-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}$$Now, I am trying to find the equation of the tangent ...
2
votes
3answers
29 views
Differentiation for a function in the integral form.
I want to know generally how we differentiate a function $F(x)$ in the following form,
$$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)...
2
votes
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 ...
1
vote
2answers
77 views
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 ...
0
votes
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(...
-3
votes
1answer
43 views
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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0answers
32 views
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?
$$\frac{dz}{dx}=z_x+z_y\...
1
vote
2answers
266 views
Optimization Problem. Find Smallest Perimeter Given Area.
QUESTION
Find the dimensions of a rectangle with area $1000$m$^2$ whose perimeter is as small as possible.
MY WORK
I think we are solving for $\frac{dy}{dx}$:
\begin{align*}
P &= (2x+2y) \\
...
0
votes
1answer
26 views
Basic Implicit Differentiation Problem
QUESTION: Find $\frac{dy}{dx}$ by implicit differentiation.
$x^2-4xy+y^2=4$
MY SOLUTION
$\frac{d}{dx}x^2 - \frac{d}{dx}4xy + \frac{d}{dx}y^2 = \frac{d}{dx}4$
$2x-(4x)'(y)+(4x)(y)'+2y\frac{dy}{dx} =...
0
votes
2answers
49 views
proof of log derivative using implicit differentiation [closed]
How would you use implicit differentiation and the fact that $\log_b x$ is the inverse of $b^x$ to prove that $$\frac {d}{dx} (\log_b x) =\frac{1}{(\ln b)x}$$
0
votes
2answers
13 views
Implicit differentiation at a point with trigonometry and a fraction.
Find $\frac{dy}{dx}$ for $xcosy-2sin(\frac{y}{2})=0$ at (2,$\frac{\pi}{3}$)
I tried using the power rule and chain rule but I do not seem to solve the problem. Can someone tell me how to solve it?.
0
votes
2answers
34 views
How to Complete Implicit Differentiation Problem
The problem I am trying to find $\frac{dy}{dx}$ of: $$\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}+\...
1
vote
0answers
17 views
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 ...
