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

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2
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4answers
112 views

Confusion about implicit differentiation.

I want to implicitly differentiate $Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$. This is not an exceedingly difficult task, and when I solved it I got $$ y' = -\frac{2Ax + Cy + D}{2By + Cx + E} $$ But my ...
1
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1answer
24 views

A strictly convex function defines an implicit function with non-positive second derivative

Problem. Let $F\colon \mathbb{R}^2 \to \mathbb R$ be a non-negative, $C^2$ function which is also strictly convex, meaning that $$ F(\lambda P + (1-\lambda)Q) < \lambda F(P) + ...
4
votes
2answers
57 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
0
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2answers
37 views

Problem understanding an implicit differentiation

Here is a general budget constraint: $p{_1}x{_1}+p_{2}x_{2}=M\Leftrightarrow \frac{p_1}{p_2}x_1+x_2=\frac{M}{p_2}\Leftrightarrow {p_{1}}'x_1+x_2=M{}'$. The main idea is that since prices are given, ...
0
votes
3answers
71 views

Find $dy/dx$ where $(7x+2y)^2=6x^4y^3$ [closed]

Find $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}$ where $$(7x+2y)^2=6x^4y^3$$ This is on my homework but book has different examples so I don't know what side to start on.
0
votes
1answer
25 views

Solving for the rate at which water is pumped into a conical tank using related rates.

Water is leaking out of an inverted conical tank at a rate of $12000.0$ cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height ...
1
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2answers
31 views

Calculus Implicit Differentiation and Concavity

Consider the relation $4x^2 - y^2 = -2$ (a) Use implicit differentiation to calculate $dy/dx$ and find all critical points of the curve. (b) Calculate the second derivative and determine the ...
0
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2answers
66 views

Differential problem, how to get y''?

I've the following equation: $b^2x^2 + a^2y^2 = a^2b^2$, the first implicit derivative is: $\dfrac{dy}{dx} = \dfrac{-b^2x}{a^2y}$ I do not undertand how to find the second derivative of this ...
0
votes
1answer
23 views

Partial Differential from Implicit Expression

I have an expression which is explicit in $p$ given by ...
0
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2answers
75 views

Weird Implicit Differentiation

While solving the sums from my textbook on implicit differentiation, I happened to encounter a sum : $$ Y^3 + XY + X^2 = 0 $$ and I was suppose to find dY/dX and substitute the value 10 for X after ...
2
votes
2answers
24 views

partial differentation of implicit function

I have a problem with this: $f(x,y,z)=\exp(xyz)$ with $g(1,1)=\ln2$ and $f(x,y,g(x,y))=2$. The task is to calculate the partial derivatives $\frac{\partial g}{\partial x}(1,1)$ and $\frac{\partial ...
2
votes
2answers
47 views

Differentiation of function

There is a passage in a physics textbook I don't quite follow. Since my question is mathematical, I've decided to post it here. The book says: Let $V$ be the volume of a molecule and assume $V = ...
0
votes
1answer
24 views

Evaluate derivatives y'(0),z'(0),y''(0),z''(0) of implicit functions y(x) and z(x)

Evaluate derivatives $y'(0)$, $z'(0)$, $y''(0)$ ,$z''(0)$ of implicit functions $y(x)$ and $z(x)$, where $y(0)=-1$ and $z(0)=1$, given by system of equations: $x+y+z=0$ and $x^2+y^2+z^2=0$ First ...
2
votes
0answers
48 views

Help solving this related rates problem.

The question: A car leaves an intersection traveling east. Its position t sec later is given by $x = t^2 + t$ ft. At the same time, another car leaves the same intersection heading north, traveling ...
0
votes
1answer
30 views

I'm having trouble with this question on derivatives.

Carlos is blowing air into a spherical soap bubble at the rate of $7 \mathrm{cm}^3/ \mathrm{sec}$. How fast is the radius of the bubble changing when the radius is $11 \mathrm{cm}$? (Round your answer ...
1
vote
2answers
54 views

Find the differential equation of all circles of radius 1 centered on the y-axis

I need to find the differential equation of all circles of the form: $$ x^2 + (y -C_1)^2 = 1$$ Differentiating w.r.t $x$ once yields: $$ x + (y-C_1) y' =0 $$ Twice: $$ 1+ (y-C_1) y'' +(y')^2 =0 $$ ...
1
vote
1answer
38 views

Implicit Function Theorem Applied

I am concerened I may have oversimplified my solution to this question. My solution: Let $F(x,y,z)=x-e^y\sin(z)$ By the implicit function theorem: $\displaystyle\frac{\partial z}{\partial ...
0
votes
1answer
42 views

Implicit function derivation

I have function $h(x,y)=e^{xy^2-1}+\log{\frac{x}{y}}-1$ and I have to find if a function $y=f(x)$ around $[1,1]$ exists. I have to check some conditions in order to find out if $y=f(x)$, ...
1
vote
1answer
53 views

Differentiation of $\cosh(xy)$

I'm doing an AP Math Course and I ran into this problem that I have never seen before and would like some help and an explanation... Consider the implicit function $\cosh(xy) = x + y$. Find ...
1
vote
2answers
34 views

implicit differentiation-trouble with algebra

I'm having trouble figuring out where to go with this implicit differentiation problem. Problem: Find $\frac{dy}{dx}$ given that $\sin(x)=e^{-y\cos(x)}$ Here is how I start: ...
1
vote
1answer
23 views

Find the equation of the normal at a point

Find the equation of the normal at the point $(2, 1)$ for the function $x^2 + y^3 - 2y = 3$ I'm still struggling a bit with the application of derivatives, but from what I understand I use the ...
0
votes
3answers
33 views

Find equation of a line tangent to a curve with a given point

Find equation of a line tangent to the curve at the given point: determine an equation of a line tangent to the curve at the given point: $9(x^2 + y^2)^2 = 100xy^2$; at the point $(1, 3)$ How would ...
0
votes
1answer
28 views

Find the slope of the tangent line to the curve.

So I am trying to find the slope of the tangent line to the curve $$\sqrt{4x+2y} + \sqrt{xy} = \sqrt{38} + \sqrt{24}.$$ at the point $(8,3)$. I ended up implicitly differentiating and getting ...
-1
votes
4answers
58 views

Second Implicit Derivative [closed]

There is a large argument over the answer to this question: Find the second implicit derivative of $x^2 + xy + y^2=2$. Can someone please answer it and explain their answer?
0
votes
2answers
69 views

Implicit function theorem problem

I have the function $$(x-2)^3y+xe^{y-1}=0$$ And I have to see if $y$ can be described as a function of $x$ around (1,1). The implicit function theorem can't be applied in this case. What should I ...
2
votes
1answer
40 views

Explain the minus sign in the following formula.

I just read that: If $z=f(x,y)=c$, be the equation of a curve, then the slope of the tangent to the curve at any point (x,y), is given by $$m=\frac {dy}{dx}=-\frac{\frac{\partial z}{\partial ...
0
votes
1answer
54 views

related rates waterskier rising off ramp

I feel this should be an easy one, but something is tripping me up. here is the question: A waterskier skis over the ramp at a speed of 30ft/s. How fast is she rising as she leaves the ramp? (the ...
0
votes
2answers
48 views

Speed of a particle given parametric equations of x and y.

So my $x(t)=\pi t +\cos(2\pi t -(\pi/2))$, and my $y(t)=\pi t +\sin(2\pi t -(\pi/2))$. I implicitly derived and got $$\frac{dy}{dt}=\frac{(\pi+\cos(2\pi t -(\pi/2))*2\pi)}{(\pi-\sin(2\pi t ...
0
votes
1answer
11 views

Implicit differentiation and tangents

is the question. I started doing it then got stuck so I looked at the mark scheme: Now, I don't understand the $-sin(-2\pi)(x\frac{dy}{dx}+y)=0$ bit $(x\frac{dy}{dx}+y)=0$ Solving ...
1
vote
1answer
31 views

Find the derived of an implicit given function.

Let $C=\{(x,y,z) \in \mathbb(R)^3| \sin x + \sin^2 y + \sin^4 z=0 \ \text{and} \ (x-z)^2=4\pi^2\}.$ By the implicit function theorem, we have that $C$ can be parametrized as a smooth curve in the ...
0
votes
2answers
36 views

Implicit differentiation of $x2^y=\ln y$

A curve has implicit equation $x2^y=\ln y$. Find an expression for $\frac{dy}{dx}$ in terms of $x$ and $y$. I got $$x 2^y \ln 2 \frac{dy}{dx}+2^y=\frac{1}{y}$$ ...
0
votes
1answer
34 views

Implicit differentiation

Find the gradient of the curve $x \ln y - \frac{x}{y}=2$ at $(-1,1)$ I've done something wrong as I got the gradient to be 0 when the answer in the back says -0.5 Can someone help me with this ...
3
votes
0answers
35 views

Applications of the quartic curve $x^2y^2-1=0$?

The quartic curve $x^2y^2-1=0$ is equivalent to the union of the hyperbolas $xy-1=0$ and $xy+1=0$, i.e., it's a rectangular hypobola superimposed with a copy of itself rotated by 90 degrees. Does this ...
2
votes
2answers
49 views

Functions that cannot be differentiated in terms of elementary functions

A while ago, I learned how to take the derivative of $y=x^x$ using implicit differentiation, and I wondered if the same trick would work on every function of this type. I tried to differentiate ...
0
votes
0answers
19 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
0
votes
1answer
20 views

Calculus Question Problem

The question states to find the equation of the tangent and normal to the curve $y^3-2y-x^3+x=15$ at (2,3). What I did was to find the gradient of that equation and ended up with ...
0
votes
1answer
23 views

Implicit Differentiation Problem

I have this problem: $\:3x^2-10x^2\ln \left(y\right)=e^{2y}$ I end up with this: $\left(6x-\left(10x^2\left(\frac{1}{y}\cdot \frac{dy}{dx}\right)+ln\left(y\right)\cdot 20x\right)=2e^{2y}\cdot ...
1
vote
2answers
77 views

How to find $\frac{d^{40}y}{dx^{40}}$, when $y= \sin x$? [closed]

What approach would be ideal in finding $\frac{d^{40}y}{dx^{40}}$, when $y= \sin x$?
0
votes
0answers
93 views

How do i show that dy/dx does not equals to zero

The question goes like this. Equation of a curve is $2x^2-3xy+y^2=5$ Find the equations of the tangent and normal to the curve at point $(4,3)$. Show that there is no point on the curve at which the ...
0
votes
2answers
68 views

How to find $\frac{dy}{dx}$ for $\sqrt{xy} = 1$? [closed]

What approach would be ideal in finding $\frac{dy}{dx}$ for $\sqrt{xy} = 1$?
-4
votes
3answers
67 views

Implicit Differentiation Solve the following using dy/dx [closed]

Solve the following using $dy/dx$. $e^x\cos(y)=xy$. then $dy/dx=?$
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0answers
79 views

What is the flaw in this proof?

Below is a proof that straight lines cannot exist in the coordinate plane. Where is the flaw in its reasoning? It will be shown that the equation of a straight line leads to a mathematical ...
1
vote
2answers
24 views

implicit differentiating equation with $\cos$

I need help getting $\frac{d^2y}{dx^2}$ for $y−\cos y=2x$ Someone answered and got $(1+\sin y(x))3+4\cos y(x)$ but i was unable to follow their steps and didnt get how to do it. any HELP?
3
votes
1answer
46 views

Implicit Differentiation Proper Answer?

I'm studying for my calc final tomorrow, and am going through some practice questions and I'm not sure if the solution is wrong or I'm just miss understanding. The question is about finding a tangent ...
0
votes
1answer
33 views

Inverse/implicit functions

How does one prove that there exists a differentiable real-valued function $y(x)$ in some neighborhood of the point $x = 0$ such that $$x = e^{-y(x)}y(x)$$ for all $x$ in that neighborhood? ...
2
votes
2answers
43 views

Implicit Differentiation usage of $\frac {dy}{dx}$

Why, after differentiating $y$ on one side of the equation, is $dy/dx$ added? As clarification an example I will provide an example: Implicitly differentiate $y^2 = x$. You get $$2y\frac {dy}{dx} = ...
0
votes
0answers
43 views

Why is $\frac {dy}{dx}$ treated as a fraction? Plus an implicit differentiation question. [duplicate]

Why is $\frac {dy}{dx}$ treated as a fraction? I always thought that it is just notation for the derivative of $y$ with respect to $x$, but when it comes to implicit differentiation and integration ...
1
vote
1answer
41 views

Need help with implicit differentiation

hi i need help on finding the $\dfrac{d^2 y}{d x^2}$ for $x^6-y^6=14$ i got $$\frac{5x^4(y^6-x^6)}{y^{11}}$$ but im not sure if its right or not also i am completly stuck on getting $\dfrac{d^2 ...
1
vote
1answer
34 views

Derivative of a minimum

The expression, $e=\left(x(t,w)-c_x\right){}^2+\left(y(t,w)-c_y\right){}^2$, has a local minimum with respect to $t$ at some $t_0(w)$. Now what does $t_0'(w)$ look like?! $x,y\in C^2$ with respect to ...
1
vote
0answers
41 views

Multiplying partial derivatives

I am trying to understand what happens when I have a continuous differentiable function $f$ on $\mathbb{R}^n$ such that $f$($x_1,x_2,...x_n$) = 0. What is the significance of the product: ...