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

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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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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
22 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 = ...
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1answer
15 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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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} ...
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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: ...
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1answer
34 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 ...
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1answer
24 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 ...
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2answers
53 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
42 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 ...
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1answer
34 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 $y=2x$. ...
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41 views

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

Is there a way to evaluate the following expression? $$ \frac{d}{dy} y' $$ where $$ y' = \frac{dy}{dx}.$$
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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
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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
27 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?
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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 ...
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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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43 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
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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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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 ...
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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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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 ...
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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
25 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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48 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
78 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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confused on a related rates problem

I'm having trouble figuring out where to begin with this problem. " a person walks along a path which is on the diameter of a circular courtyard. A light is fastened on the wall at the mid point of ...
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1answer
20 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 ...
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1answer
21 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
34 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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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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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
64 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 ...
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1answer
35 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 ...
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1answer
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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 ...
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1answer
64 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 ...
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101 views

Related Rates Problem involving two runners on a circular path

Problem: There was a typo in the original statement. I fixed it now!! Two runners start running (from the same point) in opposite directions along a circular path of radius $100\ m$ at a speed of ...
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1answer
22 views

Implicit Diff. check

I have the expression ${\dfrac{x^2+y^2}{x+y}}=3$ and I wanna find $dy/dx$. Here's my approach: $x^2+y^2=3x+3y$ $\implies (x^2-3x)+(y^2-3y)=0$ $\implies (2x-3)+\dfrac{dy}{dx}(2y-3)=0$ ${\implies ...
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Implicit differentiation involving exponential functions [closed]

How do I find $dy/dx$ for $xe^{-y/2} + ye^{-x/2} - 2 = 0$?
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Implicit Differentiation with Multiple Variables

One of my tutees is having a problem with the online portion of their homework again. I'll post the question verbatim and then explain the problem we're having. The question: Let $P = ...
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1answer
76 views

Beginner exercise on the Implicit function theorem

My first exercise on the Implicit function theorem: Show that the non-linear equation $x^4 + e^y + sin(z) + z^5 = 1$ has a "local resolution function" $(x,y) \rightarrow z(x,y)$ (is this the right ...
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Why in this derivation we have ${1\over{\cos y}}={1\over{(1-x^2)^{1/{2}}}}$?

Let $\sin^{-1}x=y$, then $\sin y = x $ and therefore: $$\eqalign{\cos y \,{dy\over{dx}} = 1&\Longrightarrow {dy\over{dx}} = {1\over{\cos y}}\\ &\Longrightarrow {dy\over{dx}} = ...
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Implicit Second Derivatives using Partial Derivatives

I've got a pretty simple derivative question for you guys. Currently, I'm a high school shop teacher preparing kids for a timed calculus competition. It's been almost 45 years since I've taken ...
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38 views

If $4x^2+5x+xy=4$ and $y(4)=-20$, find $y'(4)$ by implicit differentiation

If $4x^2+5x+xy=4$ and $y(4)=-20$, find $y'(4)$ by implicit differentiation. I implicitly differentiated the equation, but I don't see how I can use $y(4)=-20$ to my advantage.
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1answer
23 views

How to visualize implicit functions

I have a task of visualizing few implicit functions. Firstly lets say I have the following function of $N$: $$\epsilon = \sqrt{\frac{8}{N}\ln \left( \frac{4(2N)^{50}}{0.05} \right)}$$ Now this is ...
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1answer
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Help with Implicit Differentiation: Finding an equation for a tangent to a given point on a curve

When working through a problem set containing Implicit Differentiation problems, I've found that I keep getting the wrong answer compared to the one listed at the back of my book. The problem is ...
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1answer
33 views

Implicit differentiation of the function defined by the equation $y^2 = x^2 + \sin(xy)$

During implicit differentiation of this problem: $y^2 = x^2 + \sin(xy)$ I cannot figure out why at this point in differentiation this: $$2y\frac{dy}{dx}= 2x + \left(\cos xy\right) ...
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2answers
24 views

Trouble finding second implicit derivative

I have trouble finding the second implicit derivative. This is the question. Find y'' in terms of x and y by implicit differentiation. $x^5 +y^5 = 2^5$ The final answer I always get is ...
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1answer
22 views

Finding the second derivative by implicit differentiation

Find $y''=y''(x)$ for the function defined by equation $xy+y^2=2x$ I got the first derivative just fine, $y'(x)= \frac{2-y(x)}{x+2y(x)}$ but I'm having trouble with finding the second derivative. ...