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

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1answer
44 views

Alternate formula for $Γ(t + 1)$

I believe that: $$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$ Would this have any application, if it has not already been discovered, which I am almost certain that it has?
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1answer
27 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
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0answers
15 views

How can we define regular curves implicitly?

Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application. What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
2
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1answer
24 views

Difficulty Understanding Implicit Differentiation

I am struggling with an Implicit Differentiation question which is as follows: $z = (7x^4)*\ln(x)4$ where $z$ and $x$ are functions of $t$. $\frac{dx}{dt} = 4$ when $x = e$. Calculate ...
3
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1answer
105 views

Clarification on Implicit Derivatives steps

I have been attempting to wrap my head around this problem for a couple days now. I've attempted numerous different iterations to try and find how the answer is derived, but I just don't see the ...
3
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3answers
45 views

Logarithmic Differentiation equation, Help!

So, I have to differentiate this via $\log$. I am still learning, so please be patient, I will try to explain everything I did. Please tell me if it is correct. ...
2
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1answer
81 views

$y^5 =(x+2)^4+(e^x)(ln y)−15$ finding $\frac{dy}{dx}$ at $(0,1)$

Unsure what to do regarding the $y^5$. Should I convert it to a $y$= function and take the $5$ root of the other side. Then differentiate? Any help would be great thanks.
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1answer
14 views

Linearization of an implicitly defined function

$f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$ Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane. My attempt: I found that ...
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2answers
38 views

Using implicit differentiation with a fraction

How do I solve this? What steps? I have been beating my head into the wall all evening. $$ x^2 + y^2 = \frac{x}{y} + 4 $$
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1answer
33 views

why does $\frac{d}{dx} log_b(x)$ not = $\frac{lnb}{x}$?

I know that $log_b(x) = \frac{lnx}{lnb}$, and that differentiating $$\frac{d}{dx}(\frac{lnx}{lnb}) = \frac{1}{lnb}\frac{d}{dx}(lnx)=\frac{1}{xlnb}$$, so where is my mistake when I do it this way: ...
3
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1answer
42 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 ...
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0answers
38 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} + ...
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3answers
59 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. ...
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1answer
58 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
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1answer
54 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 $3/4$?
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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 ...
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0answers
27 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
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1answer
28 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 $$ z=ax + yf(a)+\phi(a) $$ ...
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0answers
34 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. ...
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0answers
22 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 ...
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0answers
48 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
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1answer
22 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
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1answer
44 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 + ...
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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 ...
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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
64 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
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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 ...
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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
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1answer
34 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 ...
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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 ...
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0answers
34 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
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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?
2
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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
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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
25 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
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1answer
35 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
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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 ...
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0answers
27 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
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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] = ...
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2answers
30 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} ...
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2answers
71 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 $(1,4)$ ...
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0answers
18 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)] &= ...
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0answers
43 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
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1answer
42 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
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0answers
25 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 ...
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2answers
59 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
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0answers
39 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
26 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
58 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 ...