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

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3
votes
1answer
31 views

Use implicit differentiation to find an equation of the tangent line to the curve at the given point

Use implicit differentiation to find an equation of the tangent line to the curve $$x^2+xy+y^2=1$$ at $(1,1)$. I am not sure how I should work this out because the given point is not on the ...
0
votes
0answers
30 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
0
votes
3answers
49 views

Struggling with integration/differentiation

quick question as I'm sure this is simple but it has me stumped. I have to integrate and differentiate this equation. Not sure on the exponential, had a couple of goes but it doesn't look right. ...
-3
votes
1answer
56 views

Confused by partial derivatives

Right, I have a question here, about the following: Use implicit differentiation to find the first and 2nd derivative. $$x^{3/5}+y^{3/5} = 7$$ The answer is: $$\begin{align}\frac{dy}{dx} &= ...
2
votes
1answer
51 views

Find $dy/dx$ of $(xy^2)+5 = x + 2y^2$

For the solution I got $$\frac{y^2-1}{ 4y-2xy} = dy/dx$$ I just want to know if this is correct. Also it says to evaluate $dy/dx$ at $(1,2)$. Would the solution to that be $3/4$?
0
votes
1answer
40 views

Related Rates problem, xy=4

$xy=4$ $a)$ Find $\dfrac{\mathrm dy}{\mathrm dt}$ when $x=8$, Given $\dfrac{\mathrm dx}{\mathrm dt} = 10$ $b)$ Find $\dfrac{\mathrm dx}{\mathrm dt}$ when $x=1$, Given $\dfrac{\mathrm ...
2
votes
0answers
25 views

Tangent line of a lemniscate at (0,0)

I need to find the tangent line of the function $y=g(x)$ implicitly defined by $(x^2+y^2)^2-2a^2(x^2-y^2)=0$ at $(0,0)$, but I don't know how. I can't use implicit differentiation and evaluate at ...
1
vote
1answer
25 views

Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.

Show $(\frac{\partial^2 z}{ \partial x \partial y})^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for the implicitly defined $z(x,y)$ as $$ z=ax + yf(a)+\phi(a) $$ ...
0
votes
0answers
33 views

Multivariable Chain Rule: Finding ∂z/∂y and ∂z/∂x

Question: The equation $$7xyz=2x^2+y^2+3z^2+7$$ implicitly defines z as a function of x and y in the neighborhood of the point where $x=2, y=1$ and $z=2$ . Find ∂z/∂x and ∂z/∂y at this point. ...
2
votes
0answers
20 views

Related Rates/Differentiation

A spherical snowball is melting in such a way that its volume is decreasing at a rate of $1~\frac{\text{cm}^3}{\text{min}}$. At what rate is the diameter decreasing when the diameter is ...
0
votes
0answers
47 views

Explain the meaning of a relationship? Vector calculus

In my vector calculus homework I have been assigned a problem that reads: Thermodynamics texts use the relationship: $$ ({dy}/{dx})(dz/dy)(dx/dz) = -1 $$ Explain the meaning of this equation and ...
2
votes
1answer
21 views

implicit function thm and chain rule

$f(x,y,z)=x^3-2y^2+z^2$, $x_0=(1,1,1)$ and $g(1,1)=1$, $f(x,y,g(x,y))=0$ find $g_x(1,1), g_y(1,1)$ using chain rule, are there any other ways to find it? my attempt at it with respect to x, I think ...
1
vote
1answer
41 views

Finding the tangent line using implicit differentiation

My professor wrote this problem on the board as a challenge: Find the tangent line at (0,0) to the curve defined implicitly below. $$\ln(1+x+y)=\left( x^{42} e^y + ...
0
votes
2answers
35 views

Implicit Differentiation involving trigonometric functions.

We are given the following condition: $$\tan(x^3y^2)=6x^2+y^2$$ Find the derivative of $y$ w.r.t. $x$, i.e., find $y'=\dfrac{\textrm{d}y}{\textrm{d}x}$ I am having trouble getting started with this ...
0
votes
0answers
34 views

How can I obtain these differential operators for this transformation?

I have transformation as the following form \begin{eqnarray} \begin{split} &u \longrightarrow \bar{u}=(ax+by+\eta)^{-3} u,\\ &x \longrightarrow \bar{x}=\frac{\alpha x+\beta ...
2
votes
2answers
61 views

Find $y'$ for $ln(x+y)=arctan(xy)$

Find $y'$ for $ln(x+y)=arctan(xy)$ Here is my attempt at a solution. Is this correct? Any hints or advice would be appreciated.
0
votes
1answer
30 views

Find the slope of the tangent line for $y^4+3y-4x^3=5x+1$ at the point P(1,-2)

Find the slope of the tangent line for $y^4+3y-4x^3=5x+1$ at the point $P(1,-2)$ Here is what I have tried. Does this look correct? Any hints or advice would be appreciated. EDIT 1: Here is ...
0
votes
3answers
42 views

Implicit differentiation q

Show that the sum of the x and y intercepts of any tangent line to the curve $x^{1/2} + y^{1/2} = 4$ is equal to 16 So far I have found the derivative, $-\frac{\sqrt{y}}{ \sqrt{x}}$, but am having ...
0
votes
1answer
32 views

Linear approximation to find partial derivatives

If the equations $f(x, y, u, v) = 0$ and $g(x, y, u, v) = 0$ can be solved for $u$ and $v$ as differentiable functions of $x$ and $y$, compute their first partial derivatives. Pretty lost on this ...
1
vote
2answers
34 views

How do you get the second order implicit differentiation?

We were given an exercise in school: Given $x^2 + 25y^2 = 100$, show that $$\frac{dy}{dx^2} = -\frac{4}{25y^3}$$ I am stuck on the first order which is $$-\frac{x}{25y}$$ When I'm now going to the ...
0
votes
0answers
30 views

Implicit function problem exercises

Let $f_1,f_2,f_3$ be continuously differentiable functions from $\mathbb R^4$ to $\mathbb R$. Give sufficient conditions so that the equations $f_1(x,y,z, t)=0$, $f_2(x,y,z, t)=0$, $f_3(x,y,z, t)=0$ ...
1
vote
1answer
52 views

Find the slope of the given curve at the point $(3,1)$

Find the slope of the given curve at the point $(3,1)$. $$2y\cos\left(\frac{\pi y}{x}\right)=2x^2-17y$$ How do I start? Differentiate and put the xy values in?
1
vote
1answer
51 views

Solving $x^2 (x-y)^2 = x^2 - y^2$ using implicit differentiation

I have a test tomorrow and I am doing homework to review and study. The problem is to differentiate $$ x^2 (x-y)^2 = x^2 - y^2. $$ I tried multiple times; however, every time I try I get the incorrect ...
1
vote
2answers
40 views

How do you get from one step to another in implicit differentiation?

So trying to understand the step process here, $$y^2 - 2x = 1 - 2y$$ So after a few simplifications we get: $$yy' - 1 = -y'$$ But what I am confused on is how that turns into $$(y+1)y' = 1$$ I ...
2
votes
1answer
23 views

Question on calculating curvature of a surface given implicitly

I want to find, as an exercise, an expression for the curvature of a surface given by the zero set of a function. I reached a final expression, but when I test it for a sphere I get a non-constant ...
0
votes
1answer
32 views

logarithmic differentiation issue

Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. $$ 1/x(x+1)(x+2) $$ and I know that $$log((ab)/c) = log(a) + log(b) - log(c)$$ So I tried to ...
1
vote
1answer
22 views

Question about closed curves and surfaces

Let $\mathbb{r}:[a,b]\to\mathbb{R}^2$ be a continuous non-intersecting loop (i.e. $\mathbb{r}(a)=\mathbb{r}(b)$ and $\mathbb{r}(x)\neq\mathbb{r}(y),\ \forall \{x,y\}\neq\{a,b\}$). Denote by $D$ the ...
0
votes
0answers
16 views

Finite Difference method for nested derivatives

I'm moderately experienced with finite difference methods, but I'm hoping that somebody has better intuition than I do. Suppose I have the following expression which I want to evaluate via finite ...
0
votes
1answer
24 views

Implicit Differentiation Solution Verification [Inverse Trig Function]

Find $\frac{dy}{dx}$. $$ \\ \\ \text{ } \\ \arctan{y^3} = \sin^3{x} + \cos^3{(yx)} \\ \text{ } \\ y^3 = \tan{(\sin^3{x} + \cos^3{(yx)})} \\ \text{ } \\ \frac{d}{dx}[y^3] = ...
0
votes
2answers
29 views

Implicit logarithmic differentiation to find the horizontal tangents of an exponential function

The graph of $y = 6{(3{x}^2)}^x$ has two horizontal tangent lines. Find equations for both of them. $$ \\ \begin{align} \\ y &= 6{(3{x}^2)}^x \\ y &= 6 \cdot {3}^x \cdot {x}^{2x} \\ \ln{y} ...
2
votes
2answers
70 views

Question on extremums, can anyone provide some help?

The real question: Let $f \colon \mathbb{R}^3 \to \mathbb{R}$; $f(x,y_1,y_2)=x^3−xy_1y_2+y_2^2−16$. Show that there exists a differentiable real function $g$ so that in some neighboorhood $(1,4)$ ...
0
votes
0answers
17 views

Implicit Differentiation Solution Verification

Find $y'$. $$ \\ \begin{align} \\5{y}^2 &= \dfrac{2x - 3}{2x + 3} \\ \\5{y}^2(2x + 3) &= 2x - 3 \\ \\ y(2x+3)&= \dfrac{2x-3}{5y} \\ \\ \frac{\partial y}{\partial x}[y(2x + 3)] &= ...
0
votes
0answers
42 views

Finding points where tangent line to the equation is horizontal

$$25x^2+16y^2+200x-160y+400=0$$ Edit: Using implicit differentiation I get: $$\frac{dy}{dx}=-\frac{25(x+4)}{16(y-5)}$$ Do I now set the numerator equal to zero? If not, what is the next step?
0
votes
1answer
38 views

Vertical Tangent line with Implicit Differentiation

The original equation is $$x^2-2xy+y^3= 4$$ and I hope the derivative to be $$\frac{dy}{dx} = \frac{2(x-y)}{2x-3y^2}$$ I know the vertical tangent is when the denominator is $0$, but I am having ...
0
votes
0answers
24 views

Explanation of differentiating implicit functions

$F(x,y,z)=0$, where $F \in C^2$ in some neighborhood of point (a,b,c) in which $F(a,b,c)\neq 0$ For the constant function $(x,y)->F(x,y,f(x,y))$ based on ...
2
votes
1answer
30 views

First-order non-linear differential equation

I have this equation: $$x(dx-dy) + y(dx+dy) = 0 $$ I tried to solve it by turning it to fraction-type: $$\frac{dy}{dx} = \frac{x+y}{x-y}$$ However, I realized that it's not homogeneous, and now I ...
-1
votes
2answers
57 views

Find dy/dx (Implicit differentation + Quotient + Trig)

I wrote question on latex and posted screenshot here, as its easier for me to type it, please see question in image
0
votes
0answers
36 views

Implicit differentiation: Differentiating function with respect to integral

I am stuck on a simple problem and would highly appreciate your opinion. I have a optimization problem over $x$ with the objective function $$F=aG(x,y)+ (1-G(x,y))(1-x)$$ So the first order condition ...
1
vote
1answer
23 views

Solving for one of variables locally when the Implicit Function Theorem does not apply

I'm having trouble deciding whether certain functions can be locally solved. I have some examples: Can $xye^{xz} - z\log y =0$ be locally solved in $(0,1,0)$ for x? y? z? In this case, I used ...
1
vote
4answers
57 views

Take the derivative of another variable

How can you take the derivative $$\frac{d}{dx}(y^2)$$ I don't understand how the chain rule applies here. Someone told me that the chain rule applies here because $y$ can be expressed in some type ...
0
votes
1answer
20 views

Linearising an implicit relation in respect to one of the variables

I have a function of this form $$ x + y = \frac{x + ay}{b^2 \sqrt{(x + ay)^2+c^2}}, $$ and I would like to find a linear approximation for $x$ in the form $$ x = y_0 + dy + \mathcal{O}(x^2), $$ what ...
0
votes
1answer
31 views

Can someone verify my derivation of a differential equation involving elliptic integrals, please?

I'm trying to determine the relationship between the major and minor radii ($a$ and $b$, respectively) of an ellipse of constant perimeter and variable eccentricity, and I've been thinking that ...
0
votes
1answer
18 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
votes
1answer
40 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 ...
-1
votes
1answer
69 views

Use implicit differentiation to find tangent lines parallel to a given line

find the tangent lines of $9x^2 + 16y^2 = 52$ that are parallel to the line $9x-8y = 1$. I have differentiated the first function to get $-\dfrac{9x}{16y}$ now I'm not sure how to relate this ...
0
votes
0answers
13 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
votes
3answers
64 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 ...
1
vote
1answer
31 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 ...
0
votes
1answer
29 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 ...
0
votes
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 ...