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

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1answer
16 views

Calculating the tangent in line given an implicitly given curve.

Given the implicitly given curve $x^4y^2-2\frac{y}{x}=ln(y-1)$ and the point P = (-1, 0). Calculate the tangent in P to the curve. I've tried to calculate the partial derivatives for x and y, and ...
3
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1answer
30 views

Finding the local extrema of a function

One of our final exam exercise sheets features this particular exercise : Find the extrema of : $2xy^3+y-x^2=0$, where $y=y(x)$ . As I thought it, this exercise involves the implicit function ...
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0answers
25 views

Applying implicit function theorem to function with derivative

This may be a very peculiar question, or I may even be on the completely wrong track, so I apologize in advance for obvious errors. I am trying to apply the implicit function theorem in an ...
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0answers
11 views

dF(x,a,b)/dy>0 iff x<g(x,a,b) then can we say that there is a upper bound on x to ensure dF/dy>0

Suppose df(x,a,b)/dy>0 iff x where g(x,y,a,b) and f(x,y,a,b) are implicit solutions to an optimization problem and x,y,a,b are parameters, then can we say that there is a upper bound on x to ensure ...
3
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3answers
57 views

Implicit Differentiation: $(x/y)+(y/x) =1$

Hi can anyone please tell me where I goes wrong with this question: Find $ \frac{dy}{dx} $ for the curves defines by this equation: \begin{align} \frac{x}{y} + \frac{y}{x} = 1 \end{align} Here ...
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1answer
26 views

Implicit function theorem application

I'm supposed to find supposed to find $ \frac{dy}{dx}$ where the function is defined implicitly as $x^2-3xy^3+x^2y^2+7=0$, by setting up $F(x,y)=x^2-3xy^3+x^2y^2+7=0$ , via direct application of ...
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1answer
27 views

Is this a valid way of computing the implicit derivative?

Suppose I wish to compute the implicit derivative of $\sqrt{x^2+y^2}=x+y$. One could differentiate both sides with respect to $x$, yielding $y\prime$ which we can make the subject. Say I were to ...
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3answers
71 views

How can I differentiate $(ye^x)^{\frac{1}{x}}=y^2$?

I have following relation to differentiate: $$(ye^x)^{\frac{1}{x}}=y^2.$$ However, I got a bit confused: I first simplified: $y^{\frac{1}{x}}e^1=y^2$ and then differentiated, but that doesn't seems ...
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6answers
596 views

When to write “$dx$” in differentiation

I'm taking differential equations right now, and the lack of fundamental knowledge in calculus is kicking my butt. In class, my professor has done several implicit differentiations. I realize that ...
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1answer
55 views

Implicit function theorem and differentiation

I am trying to do this exercise on Real Analysis topics. It's about implicit function and I don't know how to conclude anything. Here it goes: "Consider the equation $x^3 + y^3 - 3axy = 1$ where $a ...
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0answers
89 views

Equation $2x^3+6xy^2+3x^2+3y^2-y=0$ defines $y$ as a function of $x$ around the origin

Show that the equation $2x^3+6xy^2+3x^2+3y^2-y=0$ defines a function $y=f(x)$ in the neighbourhood of $(0,0)$. Determine $\dfrac{f(x)}{x}$ and $\dfrac{f(x)}{x^2}$ as $x\to0$. Any leads?
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0answers
18 views

A conceptual question on implicit partial derivatives in maltivariate situations

Say q is a function of two variables m and H: $$q = f_{1}(m,H)$$ Further, m and H are related to a third variable L: $$L = f_{2}(m,H)$$ I believe that, using the idea of the total derivative, it is ...
2
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1answer
35 views

derivative of implicit function

supposed I have a function which is $q_2f_2(δq_1+q_2)$ I want to know the second derivative of the function w.r.t $q_2$ Firstly, I took the first derivative w.r.t to $q_2$ and I got the result as ...
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2answers
44 views

Taking derivatives of the implied function - from the implicit function theorem,

I showed that the relation $$f(x,y)=e^x - e^y + xy = 0$$ defines near (0,0) an implicit function y=$\phi (x)$, since the $1x1$ block, $\frac{df}{dy}$, evaluated at (0,0) gives -1, which is non-zero - ...
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2answers
28 views

Implicit differentiation

I am currently working on a question which involves me differentiating $$\frac{y}{x}$$ I can't find nothing in books or on the internet about how to deal with this kind of implicit differentiation. ...
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2answers
40 views

implicitly differentiating polar equations

For polar coordinates, we have the following equations. $x^2 + y^2 = r^2 $, $x= r \cos(\theta) $, and $y= r \sin(\theta)$. When I find $ \frac {\partial r}{\partial x}$, I have the following: ...
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1answer
24 views

finding largest value of x obtained from curve by implicit differentiation

Consider the curve defined by $x^2 + 2y^2 + 4 \beta x y = K $ with $K > 0$ and where $\beta$ is a (sufficiently small) parameter. Assuming that the above can be used to define a function $x = ...
0
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1answer
16 views

deriving formula for $y'$ in terms of various partial derivatives

Consider two three-variable functions $H(x, y, z)$ and $K(x, y, z)$ and the associated level surfaces $H(x, y, z)= a$ and $K(x, y, z)= b.$ It is assumed that these surfaces intersect along some curve ...
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2answers
23 views

difficulty proving this theorem

Let $x = s + t$ and $y= s - t$. For any $f(x, y)$, let us define this function in terms of s and t in the usual fashion: $g(s, t): = f(x(s, t), y(s, t))$. Show that $$ \left(\frac {\partial f} ...
0
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1answer
25 views

If $f(x,y)=\sin^{-1}\left(\frac x y\right)$ determine $\left(\frac{\partial f}{\partial x}\right)_y$

I have attempted the question but I am not sure whether I am right. Here's what I have done: $$sin(f)=\frac x y$$ Differentiating implicitly with respect to $x$, holding $y$ constant: ...
0
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1answer
36 views

Implicit differentiation, what am I doing wrong?

The problem is as follows: $y^2(2-x)=x^3$ I know now how to solve it. However, my first try is described underneath, and I still do not know why that did not work. Could anyone explain it to me. I ...
0
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1answer
26 views

Limit of Implicitly Defined Function (Follow up)

Is there a method such that one can determine the limit $\lim_{h \to 0}\frac{f(x)}{x^n}, n \in \mathbb{Z}^{+}$ for an implicitly defined function, defined near the origin - such as ...
2
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2answers
58 views

Inconsistent answers when implicitly differentiating polar identities

Currently doing a problem where I need to find $\frac {\partial \theta}{\partial x}$. However, for $\tan(\theta)= \dfrac yx$, $\frac {\partial\theta}{\partial x}$ is yielding $- ...
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2answers
44 views

Using Differentials Problem: Can't Separate x and y

I've been asked to estimate a y coordinate by using differentials. This normally isn't overly difficult, however, I'm not sure what to do in a case like this when y cannot be separated and used as a ...
2
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1answer
35 views

Further differentiation

The question tells us that the functions $x(t)$ and $y(t)$ satisfy $dx/dt=-x^2y$ and $dy/dt= -xy^2$ when $t=0$, $x=1$, and $y=2$. I have already worked out that $dy/dx= (xy^2)/(x^2y)$ and that $y=2x$. ...
3
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1answer
43 views

How to calculate $\frac{d}{d y} y'$

Is there a way to evaluate the following expression? $$ \frac{d}{dy} y' $$ where $$ y' = \frac{dy}{dx}.$$
2
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2answers
48 views

Why is Implicit Differentiation needed for Derivative of y = arcsin (2x+1)?

my function is: $y = arcsin (2x+1)$ and I want to find its derivative. My approach was to apply the chain rule: ${y}' = \frac{dg}{du} \frac{du}{dx}$ with $g = arcsin(u)$ and $u = 2x+1$. ${g}' = ...
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1answer
14 views

Help with Related Rates problem

I'm slowly working my way through a ton of these problems but have come across one that has me stumped. Here's the problem in full: Water is leaking out of an inverted conical tank at a rate of ...
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1answer
30 views

How to solve implicit differentiation with radicals?

Here is the question: $$\sqrt{y^2\sin^2x + x^2\cos^2 x} = 4xy$$ I know about product rule and such but I'm exactly sure how to begin. edit: I'm just trying to get the first derivative of this ...
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4answers
58 views

Differentiate $\tan(xy)= y+2$

Here is what I did: $$\tan(xy)=y+2$$ $$(xy')(y)\sec^2(xy)=y'$$ Now I'm stuck on simplifying this. How do I get all the y's on one side and divide?
3
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2answers
70 views

Implicit differentiation: $\ln(1+xy) = xy$

$\ln(1+xy) = xy$ When I try to implicitly differentiate this I get $\frac{1}{1+xy}(y + xy')$ = (y + xy') At which point the $(y + xy')$ terms cancel out, leaving no $y'$ to solve for. However, the ...
3
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0answers
47 views

Implicit differentiation gives different results

I want to find an expression for the derivative of the implicit function below: $$\arctan(xy)=\frac{\pi}{4}e^{x-y}, \qquad \text{at point } (1,1)$$ I've tried to derive this using both Maple and ...
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2answers
44 views

How to differentiate $y=\frac{2^x+4^x}{3^x+5^x}$

Differentiate $$y=\dfrac{2^x+4^x}{3^x+5^x}$$ I think you have to use implicit differentiation, but I don't know how to start. I first ln both sides and separated the fraction into ...
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1answer
17 views

Help with simplifying implicit differentiation

Given the equation $\frac{y}{x+7y} = x^6 + 7$, find $\frac{dy}{dx}$. Ok, so I started to solve for $\frac{dy}{dx}$ and got to here: $\frac{\frac{dy}{dx}(x+7y)-(1+7\frac{dy}{dx})(y)}{(x+7y)^2} = ...
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0answers
20 views

How does a Function solving a Functional Equation changes with respect to a change of a Parameter of that Equation?

I want to see how a function solving a functional equation changes with respect to a change a parameter of the functional equation. In particular, let $C(X)$ be a Banach space with continuous and ...
3
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1answer
36 views

How do I Implicitly Differentiate this equation?

My equation is $y=x^{y^2}$ I did the $\ln$ of both sides, then I tried implicit differentiation. I got $$y'= \frac{x^{y^2} y^2}{x}.$$
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3answers
33 views

Use implicit differentiation to find the largest y-value in the loop of the Folium of Descartes, which is given by :$ x^3 + y^3 - 3xy = 0$.

My professor didn't even explain how to do this. I asked the TA,and they told me to equal the first derivative to $0$ and solve for $x$ and then see if the second derivative is less than $0$? Can ...
3
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6answers
70 views

Derivative of $y=x^{\ln x}$?

I only know how to do one step: $$ \ln\left(\,y\,\right) = \ln\left(\, x^{\ln\left(\, x\,\right)}\,\right) $$ how do i do the derivative of $\ln\left(\, x^{\ln\left(\, x\,\right)}\,\right)$ ?. I know ...
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1answer
30 views

Taylor series of an implicit function

Suppose the function $s:[-\delta, \delta] \to \mathbb{R}$, $\delta > 0$, is defined implicitly by $$s(t) = 1 - c\beta t (s(t))^{\beta}$$ for some $c > 0$, $0 <\beta < 1$. Can an ...
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3answers
52 views

Property of Implicit Function - Calculus

The differentiable fuction $z=z(x,y)$ is given implicitly by equation $f(\frac{x}{y},z)=0$, where $f(u,v)$ is supposed to be differentiable and $\frac{\partial f}{\partial v}(u,v)\neq0$. Verify that ...
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1answer
118 views

Second derivative of x^(4) + y^(4) = 16 by implicit differentiation

Find $y''$ if $x^4 + y^4 = 16$ by implicit differentiation So after the first implicit differentiation I got this equation (let's call it A): $4x^3 + 4y^3*\frac{dy}{dx} = 0$ Where $\frac{dy}{dx}$ is ...
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1answer
21 views

Differentiation to Find slope if tangent line Implicitly

I have the equation $x^3+y^3-4xy=8$, I need to find the equation for the tangent line at $(2,0)$. When I derived the equation I came up with $y'=(3x^2-4x)/(-3y^2-4y)$ (sorry I don't know how to format ...
0
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1answer
24 views

On which points of $xy=(1-x-y)^2$ is the tangent parallel to the $x$-axis?

On which points of $xy=(1-x-y)^2$ is the tangent parallel to the $x$-axis? All I get is the derivative of the function, as far I know, I set the derivative equals to zero.
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1answer
36 views

Implicit differentiation: Describing where a graph is increasing or decreasing

Considering $s$ is implicit to function of $p$, given by $s^6 - p^4 = 1$. For what $s$ is it increasing and decreasing? Well, I answered first like following: Calculating the first derivative using ...
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2answers
30 views

How can I differentiate correctly in this problem so that the units work out correctly?

I have this homework problem: Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is ...
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1answer
38 views

Proof of Multivariable Implicit Differentiation Formula

If the equation $F(x,y,z)=0$ defines $z$ implicitly as a differentiable function of x and y, then by taking a partial derivative with respect to one of the independent variables (in this case x), you ...
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3answers
65 views

Differentiate:$ x^{2y} = \ln y$

I am having trouble with the following equation defined implicitly with respect to $x$.I figured I could use the quotient rule or maybe play around with the logs, however I always need to ...
0
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0answers
47 views

implicit differentiation simplification

given that $y'=\frac{tan(y)}{1-xsec^2(y)}$ find $y''$. I've definitely differentiated correctly, I've checked it many times but I can't simplify to what wolfram alpha gets, namely ...
0
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1answer
19 views

Use implicit differentiation to find an equation of the tangent line to the curve at the given point (2,4)

Find an equation of the tangent line to the curve at the given point (2,4) $$x^2+2xy-y^2+x=6$$ I got a derivative of (-2y-1-2x)/(2(y-x)) and a slope of $-3.25$ and $y=10.5-3.25x$ as my equation of ...
0
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1answer
89 views

Related Rates Problem about moving shadow

I have another question about related rates. I have been asked the following question about related rates. It's been a while since I looked at related rates. I appreciate if anyone can help me with ...