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

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13 views

Generalize Implicit Differentiation to find Tangent Plane

For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ ...
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3answers
27 views

Differential equations stuck [closed]

How to solve $y'+y=4x$ I dont know where to begin.
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1answer
70 views

What's the arc length of an implicit function?

While an explicit function $y(x)$'s arc length $s$ is easily obtained as $$s = \int \sqrt{1+|y'(x)|^2}\,dx,$$ is there any formula for implicit functions given by $f(x,y) = 0$? One can use the ...
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33 views

Ordinary differential equations as variational problems

Considering an ordinary differential equation of first order in the implicit form $$ F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0 $$ with $\alpha,\, q_0$ constants, what is the relation of the solution ...
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2answers
34 views

Find the derivative $y'_x$ from the equation $y^3 + x^2 = xe^{y^2} - y\sin x$

Find the derivative $y'_x$ (edit: $y'_x = {dy \over dx}$) from the equation $$y^3 + x^2 = xe^{y^2} - y\sin x$$ I generally don't understand how it should be done. Should I just implicitly ...
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1answer
25 views

Calculation of Rise in Height of water in a Frustum of Right Circular Cone

A volume of frustum of right circular cone is calculated as follows. With known h, R & r of a container with the shape shown below, how to find out the rise in height for each time $7m^3$ of water ...
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0answers
28 views

partial deravtive of $\sin xyz - { 1 \over z-xy} = 1$

Given z(x,y) and $\sin xyz - { 1 \over z-xy} = 1$. How to calculate $z_x(0,1)$ ? Let $$F=\sin xyz - { 1 \over z-xy} - 1 = 0 $$ $$z_x= - \frac{F_x}{F_z}= -{y(z+xz_x) \cos xyz + \frac{z_x - y}{(z-xy)^2} ...
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4answers
92 views

Second derivative of $x^3+y^3=1$ using implicit differentiation

I need to find the $D_x^2y$ of $x^3+y^3=1$ using implicit differentiation So, $$ x^3 + y^3 =1 \\ 3x^2+3y^2 \cdot D_xy = 0 \\ 3y^2 \cdot D_xy= -3x^2 \\ D_xy = - {x^2 \over y^2} $$ Now I need to ...
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4answers
39 views

Implicit Differentiation in multivariate calculus

Let $y(x)$ be the be given explicitly by the equation : $xy\left( x\right) -\ln y\left( x\right) = 1$ Determine $\dfrac {dy}{dx}$ I'm unsure of how to go about this problem.
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1answer
12 views

Classifying points found with derivative

I've been trying to learn derivatives recently. I have troubles with finding local minimums and maximums given an equation I have understood this far $y=2x^3-9x^2+12x-5$ $dy/dx= 6x^2-18x+12$ $x=1$ ...
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2answers
26 views

Find x coordinates of the horizontal tangent line. [closed]

This is the tangent line of a function. How can I find all the x coordinates of the line tangent to the original function? $$y'={2x\over3y^2 +2y-5}$$
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1answer
34 views

Find the point on the graph of $g(x) = 2x^2 + 3x + 1$ at which the normal line is parallel to the line with equation $x - y = 2$

I suspect I should use implicit differentiation at some point to solve this problem but I'm somewhat clueless on how to approach this problem. Would appreciate tips and directions
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1answer
46 views

Why can't one implicitly differentiate these two relations?

This is from Tenenbaum & Pollard's "Ordinary Differential Equations" book, chapter 1, Exercise 2, problem # 16. There are two relations introduced in problems 14 and 15: (1) $\sqrt{x^2 -y^2} + ...
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2answers
38 views

Separating differential equatons

The initial equation: $$ y''= g-((C*(y')^2)/m) $$ and I am trying to separate it into two differential equations. I also have that the aerodynamic force $F=C*(y')^2$. The initial equation describes ...
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3answers
107 views

the derivative of $ {1\over x} + {1\over y} = 1$

I am finding the derivative of this equation, using the implicit differentiation in term of x. $$ {1\over x} + {1\over y} = 1$$ Here is what I did. $$ {1\over x} + {1\over y} = 1$$ $$ x^{-1} + ...
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2answers
65 views

implicit differentiation: some misconceptions

If we had an equation like $xy+xy^3=2$, I know how to get $y'$. The question is does the RHS always equals to the LHS or for certain values of $x$ only? I am so confused about this part. Does $y$ in ...
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0answers
18 views

Determine and classify the local extreme values of this equation using the second derivative test

$F(x,y)= x^5-3x^2-2y$ I've found the first and second partial derivatives of this function but $F_{yy}(x,y)=0$ and $F_{xy}(x,y)=0$ so when I plug this into the equation $D$, it equals $0$. How do I ...
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0answers
18 views

Implicit Solution to a differential equation

I'm looking at the ODE: $\frac{dY}{dX} - \frac{ X^2 + 2 Y^2 - 1 }{ ( Y - 2 X )X } = 0$ I'm looking for an $implicit$ solution to the above. Meaning, I want to find a relation $F(X,Y)=0$, where ...
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1answer
29 views

Deriving the Formula of Total Derivative for Multivariate Functions

How is the formula for the total derivative with respect to x derived? The formula is: $$\large \frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dy}{dx}+\frac{\partial ...
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1answer
29 views

Found the second derivative of an implicit function

Let $f(x,y)=2x^3-4xy+2y^3$ knowing that $y=f(x)$, use the implicit function theorem to find $y''(1)$ So, if $x_0=1$, then $f(1,y_0)=2-4y_0+2y_0^3$ and $f(1,y_0)=0$ in 3 points. Lets take ...
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0answers
25 views

How do I prove the result… [duplicate]

http://i.stack.imgur.com/1Smau.jpg How do I proceed in order to prove the above result. help needed ...thanks in advance
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17 views

Proving a $f(x,y)=x\cos(xy)$ has one solution $y(x)$ around $(1,{\pi\over 2})$, using the Implicit Function Theorem.

I did show that that function is $C^1$ and that $f'_y(1,{\pi})$ is invertible, but I don't understand how the theorem so directly shows uniqueness, and therefore don't know how to apply it. I would ...
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0answers
15 views

Implicit function theorem, what is the meaning of invertible linear operator?

I have to show that around $(1,-1,0)$(have to find the neighborhood as well), $x,y$ are determined uniquely by $z$ given $x+yz-z^3=1, x^3-xz+y^3=0$. What I did so far is: $f:\Bbb{R}^2\times\Bbb{R}\to ...
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0answers
12 views

Pole part of gamma function

In the pole expansion of gamma function, I hope to differentiate the pole part and the regular part with respect to $\alpha$ and to see the sum of the two derivatives vanishes. This shows that the ...
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1answer
19 views

i have a few questions regarding differentiation using the first principle

Differentiate the following using the first principle $$f(x)=x^\frac{3}{2}$$ $$f(x)=\frac{1}{\sqrt x}$$ i have tried the questions but i am stuck in the second step
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1answer
15 views

Differentiation of a function with respect to an extra variable

I was looking at the following answer to a related rates problem, and it got me thinking: How is that the variable $t$ comes up in the bottom differential? If I multiply both sides by $dt$ and then ...
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2answers
66 views

Implicit differentiation to find property of a function

Im asked to show that if $h(1)=0$ and $h'(x)={1\over{x}}$ then $a,b>0$ show $h(ab)=h(a)+h(b)$. Im expected to use implicit differentiation to show this property.
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1answer
36 views

Implicit differentiation to find derivatives of a function whose only defined by its derivative

A question I have asks if $K(x)$ satisfies $K(1)=0$ and $K'(x)={1\over{x}}$ then show: If $f(x)=K(10x)$ then $f'(x)={1\over{x}}$ So Im not sure if I have to prove it or do something else but this is ...
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0answers
55 views

Partial Derivatives Calculus

If $$x^x\cdot y^y\cdot z^z=c$$ then prove that $$\frac{\partial^2z}{\partial x \, \partial y}=(-x\log_ex)^{-1}$$ How to do this? I was first taking logarithm and then differentiating but did not work. ...
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29 views

Solve for saddle point

let $f(x,y,z)=z^3+xz+xy-x$ a. show that there exists a neighbourhood of $(0, 0, 1)$ where $z = g(x,y)$ b. show that $g$ has a saddle point at (0, 0) c. approximate g(0.1, -0.1) while part a is ...
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1answer
14 views

Find the equation of the line for this impilcit differential

Here is the question. A set of points in this graph that satisfies the the equation of the line tangent to this curve at the point (0,4) So I started by finding the derivative But I am not ...
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1answer
34 views

Points on a normal to a superellipse at distance $d$ from the curve

Given a point P0 $(x0, y0)$ lying on a super ellipse, $(x/a)^n + (y/b)^n = 1$, where $2 <= n <= 5$, I'm trying to derive an equation to describe the point P1 (x1, y1) lying on the normal through ...
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2answers
29 views

Few questions on implicit function [duplicate]

When we trying to differentiate a function $y=f(x)$, we are actually finding the rate of change of $y$. But what do we exactly mean by differentiating both sides of the equation, say $x^2+y^2=1$, with ...
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32 views

How would I graph this implicit function in geogebra?

I am trying to graph the below equation in geogebra. \begin{equation} x\cos { (xy) } = 4 -y \end{equation} I was able to get wolframalpha to graph it but I am unsure on how to in geogebra. I am ...
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1answer
21 views

Conceptual problem concerning differentiation of implicit functions

I have the following question concerning the implicit function theorem (and, I think, more generally the behavior of a differential operator, such as the one used for derivatives). [I hope the ...
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1answer
22 views

$x^2-5xy^3-2y^4+3=0 , y\geq0$ find $f'(0)$

I need get value $$f'(0), y=f(x)$$ $$x^2-5xy^3-2y^4+3=0 , y\geq0$$ How I can differentiate when there is $y$? I found some examples but I am little bit confused.
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1answer
23 views

Proving an implicit function is strictly decreasing

I'm trying to prove from the following implicit equation: $$ \frac{\sin (p)-\sin (a)}{p-a}=\cos (p)\land -\frac{\pi }{2}\leq a<\pi \land \pi <p\leq \frac{3 \pi }{2} $$ that $\cos(p)$ is ...
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1answer
24 views

Taking derivatives of constants and variables

When implicitly differentiate a function, for example, $f(x)=(G)(x)$, where G is a constant, is it possible to differentiate it such that we can treat G as a variable? From my understanding it is ...
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1answer
19 views

Find the partials $\frac{\partial z} {\partial x}$ and $\frac{\partial z}{\partial x}$ with $z$ defined implicitly as a function of $x$ and $y$

Find the partials $\frac{\partial z} {\partial x}$ and $\frac{\partial z}{\partial y}$ with $z$ defined implicitly as a function of $x$ and $y$ The equation is $3x^2z-x^2y^2+2z^3+3yz=5$ I'm ...
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0answers
32 views

Implicit Differentiation with partial derivatives.

Given that: $xs^2+yt^2=1$ and $x^2s+y^2t=xy-4$ where $x=x(s,t)$ and $y=y(s,t)$, find $\dfrac{\partial x}{\partial s}$, $\dfrac{\partial y}{\partial s}$, $\dfrac{\partial x}{\partial t}$ and ...
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2answers
32 views

implicit differentiation computation

I am having troubles isolating and solving for $\frac{dy}{dx}$, $\sin(x+y)=xy$ $\cos(x+y)(1+\frac{dy}{dx})= x\frac{dy}{dx} + y$ What would be the next few step to solve for $\frac{dy}{dx}$?
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1answer
71 views

Are all solutions of $ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$ periodic?

Let $n$ be a strict positive integer and $x$ is a complex number. Define $f_n(x)$ as one of the solutions to $$ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$$ Where the derivative is with respect to $x$. Why ...
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0answers
18 views

Finding parameter from inverse relations

$ x,y,z$ are known functions of $t$ and $f$ is a known function: $$ z(t)= f ( x(t),y(t)) ; $$ In order to compute $t$ as a function of $x$(or other dependent variables) how to define some ...
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4answers
413 views

How is implicit differentiation formally defined?

I get that differentiation is an operation used on a function, so if a function is defined $x\mapsto x^2$, the derivative is $$ (x\mapsto x^2)' = x \mapsto \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h} ...
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4answers
67 views

Derivative of $4x^{5x}$

I'm studying for an exam and I'm confused on these type of derivative problems. I know the answer I'm just confused as how to get to the answer. Would anyone mind going through the steps: Question: ...
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2answers
33 views

Derivative of $\ln(xy+1)=\sin(\pi x)$ at P(1,0) using implicit differentiation

Firstly, I confirmed P(1,0) is on the curve by substitution. Then I differentiated both sides giving me $\frac{x \frac {dy}{dx}+y}{xy+1}=\cos(\pi x)$ So $\frac {dy}{dx}=\frac {\cos(\pi ...
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1answer
81 views

First and Second derivatives differentiation

The equation $x^5y + x +y^3 =3$ defines implicitly a function $y=g(x)$ near $x=1$. Compute $g(1)$ , $g'(1)$, and $g''(1)$. If someone could show me the first few steps that would help.
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3answers
76 views

dy/dx implicity

Question: Find $\frac{dy}{dx}$ implicity if $x=\tan(xy)$. So far, I have used implicit differentiation on both sides of the equation and ended up with two different answers. The first one was a bit ...
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3answers
62 views

How do you implicitly differentiate $y$ from $y\sqrt{x^2+y^2} = 15$?

I've been working on this problem for the last 45 minutes, and I keep getting the wrong answer, no matter what I do. I tried squaring the whole equation, so that there was no radical to deal with - ...
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1answer
41 views

Implicit differentiation with 3 variables and 2 simultaneous equations

Translated from Danish: The "equation system" $e^x+y^2+x=2\,\, \land\,\, z^2+xy=4$ implicitly defines $x$ and $z$ as functions $x=x(y)$ and $z=z(y)$ of the variable $y$ around the point ...