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

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Numerical Methods For Implicit Differentiation

I am implementing a Genetic Programming algorithm that uses a special cost function to perform Symbolic Regression (see this paper) (essentially curve fitting, but instead of finding parameters/...
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4answers
61 views

Prove using induction the following equation is true.

If $$(1-x^2)\frac{dy}{dx} - xy - 1 = 0$$ Using induction prove the following for any positive integer n$$(1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+3)x\frac{d^{n+1}y}{dx^{n+1}} - (n+1)^2\frac{d^ny}{dx^n} ...
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2answers
27 views

Differential $\mathrm{d}^2f$ of implicit function $F(x,y,z)=xyz-x-y-z=0$

Determine the differential $\mathrm{d}^2f$ of the implicit function defined as $z=f(x,y)$: $$F(x,y,z)=xyz-x-y-z=0$$ So in fact of the implicit function I have to use the implicit function ...
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1answer
26 views

How to solve this implicit differentiation problem concerning arcsin?

My overarching question is about differentiating when you have these inverse trig functions, but listed below is the specific question I am trying to solve. If you help me with the problem, it'll help ...
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3answers
58 views

Is this derivation of implicit differentiation using chain rule is technically valid?

In my text book I found a derivation of implicit differentiation using the chain rule to get this formula: $$\frac{dy}{dx}=\frac{-\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$$ ...
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1answer
98 views

What is the derivative of $x^{x^{x^{x^{.^{.^{.}}}}}}$ [duplicate]

Here is my attempt: Substituting y for infinite x powers: $$x^{x^{x^{x^{.^{.^{.}}}}}}=y → x^y=y $$ Giving: $$x=y^{\frac{1}{y}}$$ Take natural logs & differentiate with respect to $y$: $$ln(x)=...
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how to find the the maximum of an implicit function

I have an implicit function and I would like to find the value of $h$ that maximizes $R$, i.e, I want to find $h$ that satisfies $\frac{\partial R}{\partial h} = 0$. The function is, $C=\frac{A}{1+\...
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1answer
30 views

Error propagation - implicit functions

I have a little problem that I should solve quickly and I'm a little bit on pressure, so that any help/tip would be of great help. I have two nonlinear equations with two unknown variables x and y ...
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25 views

How to find the maximum of an implicit function in matlab

I have an implicit function with 2 variables x(1) and x(2). The variable x(2) can not be written explicitly in terms of x(1). I would like to find the value of x(1) that maximizes x(2). The implicit ...
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2answers
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Partial Derivative of implicit function z defined as a function of x and y

Suppose that z is defined implicitly as a function of x and y by the equation $ x^2 + yz ...
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1answer
24 views

Asymptote of non-algebraic implicit function

The function f is implicitly given by $g(x,y)=e^{xy}-x-y=0$. I want to know about the asymptotes towards $+\infty$. I could deduce that there is only one intersect with the x-axis at (1|0) and by the ...
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1answer
33 views

Implicit differentiation of a two variables function

A function $z=z(x,y)$ is given implicitly by the function $$f\left(\frac{x}{y},\frac{z}{x^{\lambda}}\right)=0$$ where $\lambda\in\mathbb{R},\lambda\neq0$. I have to show that if $f(u,v)$ is ...
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6answers
109 views

find ${dy}/{dx}$ if $x^y + y^x = 1$

Find ${dy}/{dx}$ if $x^y + y^x = 1$. I have no idea how to approach this problem. Can somebody please explain this to me?
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2answers
46 views

Finding equation of tangent lines to hyperbola $xy=1$

I am working through some calculus problems (this is in a section on implicit differentiation) and this one is giving me trouble. I am trying to find the equations of the tangent lines to the ...
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1answer
51 views

For $f_{a,b}(x)=2^{x+a-b}-abx+4$, show that for $x_1,x_2$ close enough to $2,3$ there are $a,b$ such that $f_{a,b}(x_1)=f_{a,b}(x_2)=0$

Let $f_{a,b}(x)=2^{x+a-b}-abx+4$. Show that for $x_1$ close enough to $2$ and $x_2$ close enough to $3$ there are $a,b$ such that $f_{a,b}(x_1)=f_{a,b}(x_2)=0$. Hint: $f_{2,2}(2)=f_{2,2}(3)=0$. It ...
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1answer
47 views

Which derivative is correct?

Consider the following expressions: $$C_{i}=\sum_{j=1}^{N_i}v_{j}, \quad v_j \in \mathbb{R}, \quad N_i \in \mathbb{N} $$ \begin{equation} x_{i}=\frac{C_{i}}{N_i} \end{equation} I want to obtain an ...
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0answers
9 views

Is this implicit mapping convex?

I am interested in the convexity properties of the following mapping on the $n\times 1$ vector $x$: $$ x_{j}=y_{j}^{\beta}\left(\sum_{i=1}^{n}B_{ij}x_{i}\right)^{\alpha} $$ where $\beta>0$, $y_j\...
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4answers
113 views

Is g with $\cosh(xg(x))=x\cosh(g(x))$ decreasing?

Let $g:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ be defined implicitly by $\cosh(xg(x))=x\cosh(g(x))$ and $g(1)\sinh(g(1))=\cosh(g(1))$. How to show that $g$ is differentiable? Furthermore, is it ...
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2answers
38 views

Finding the normal to a curve

Find the equation of the normal to the curve with equation $4x^2+xy^2-3y^3=56$ at the point $(-5,2)$. I know that the normal to a curve is $$-\frac{1}{f'(x)}$$ And when I differentiate the curve ...
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0answers
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Generalised gradient of implicitly defined function

In the textbook 'Optimization and nonsmooth analysis' Clarke talks about the generalised gradient of a Lipschitz function, akin to the derivative of a differentiable function. He also shows that the ...
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1answer
36 views

Implicit function theorem, beginner question

I'm revising things for the end-term semester exams, and Implicit Function Theorem is something that we studied for 1 month (theoretically), with every possible alternation on the theorem and it's ...
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Related rates of $y^2-3xy+x^2=25$ given $x$ and $y$ are dependent of $t$

I am given the following problem: Two variables $x$ and $y$ are dependent of $t$ and are related according to $$y^2-3xy+x^2=25$$ If $x$ varies $1$ unit when $x = 0$ then find $\frac{dy}...
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3answers
35 views

Implicit differentiation of Product of Two Functions of $y$

I am asked to find the derivative of $$e^y\cos^2y$$ with respect to $x$. I think it is $$y'e^y\cdot2y'\cos y \sin y$$ Since there is no $x$ and $y$ term, such as $xy$, the product rule does not apply(...
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Differential equations with inverse trigonometrical functions

$(x^2+y^2)^(1/2)$=$e^(asin(y/(x^2+y^2)^1/2))$ Prove that $\frac{d^2 y}{dx^2}$=$2((x^2+y^2))/(x-y)^3$, x>0 I started with taking the natural log of the given equation and differentiating it, I ended ...
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2answers
90 views

Solution of partial differential equation

Solve the differential equation, $$ z=\frac{\partial z}{\partial x}x + \frac{\partial z}{\partial y}y+ (\frac{\partial z}{\partial x})^2 + \frac{\partial z}{\partial x}\frac{\partial z}{\partial y}+ (...
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1answer
33 views

Two methods of implicit differentiation don't correspond

I recently attempted a question on implicit differentiation twice. I differentiated using one method in the first attempt and then another method in the second attempt but they do not correspond when ...
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3answers
53 views

I did a Maths question on mechanics using calculus and physics students disagreed, why are the solutions different? [duplicate]

So, I did the following question and got 2 different answers. Question: A lighthouse located 300 metres from a straight shoreline sweeps its beam of light around in a circle at a constant rate of 1 ...
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28 views

Using Implicit Differientation to find a formula for dy/dx

I already have all the steps down, one thing I couldn't understand was how to get the final answer (which I have also): The equation given is $y=xe^y$ Here is the steps I've taken so far: $\frac{...
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0answers
27 views

Implicit function theorem, sum of derivatives

Because of the implicit function theorem, is it in general true that: $$ \frac{\partial f}{\partial x} = \frac{\partial f}{\partial a} \frac{\partial a}{\partial x} + \frac{\partial f}{\partial b} ...
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2answers
36 views

implicit derivative of e^y

I am confused about this problem of finding the derivative of $e^y$ when differentiating with respect to x. The whole problem is to differentiate $y = x \, e^y$ with respect to $x$ but I get stuck on $...
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1answer
36 views

Consider the curve with implicit equation…

I have the last question on my assignment which I cannot answer and was wondering if any of you could help me. The two part question reads Consider the curve with implicit equation $x^\frac{2}{3} + ...
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2answers
39 views

Differentiating Query

Does the following make logical mathematical sense: $$x^2=t$$ $$\frac{d} {dy} (x^2)=\frac{d} {dy} (t)$$ $$2x\cdot\frac{dx}{dy}=\frac{dt} {dy} $$ $\mathbf{\therefore \frac{dy} {dx} =2x \cdot\frac{dy}{...
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1answer
41 views

Basic Implicit Differentiation

The curve C has equation $2x^2 + y^2 =18$. Determine the coordinates of the four points on C at which the normal passes through the point $(1, 0)$. Here's what I did: And, $m_{normal} = \frac{...
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3answers
35 views

How to find the slope of curves at origin if the derivative becomes indeterminate

What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg-- What is the slope of the curve $x^3 + y^3= 3axy$ at origin and how ...
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2answers
35 views

Relation between chain rule and implicit differentiation derivation in multi variable calculus

So my question is on the derivation of the implicit differentiation (taken from here). The general chain rule, from here, it says that if we have a function $z$ of $n$ variables, $x_1, x_2,\ldots,x_n$...
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4answers
43 views

Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$

I was solving a few problems from a textbook and I came across this one: Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$ I could find the tangency ...
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2answers
33 views

Implicit derivation to find $\partial x/\partial v$?

I saw this question: $$\begin{cases} x^2+y^2=u \\ x\sin y+y=v\end{cases}$$ What is the $\partial x/\partial v$? I think it should be $1/\sin(y)$ because $\partial v/\partial x=\sin y$, but the ...
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2answers
33 views

Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
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3answers
44 views

Implicit Differentiation - What am I doing wrong?

I need to find $y'$for the following equation: $$ e^{\frac{x}{y}} = x-y $$ Before differentiating I decided to perform a quick rewrite: $$ \begin{align*} e^{\frac{x}{y}} &= x-y \newline \...
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1answer
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What is the second derivative of $Tr(A^T(\alpha)BA(\alpha))$?

What is the second derivative $\frac{d^2}{d\alpha}Tr(A^T(\alpha)BA(\alpha))$? Here, $B$ is square matrix and $A(\alpha)$ is a parameter dependent matrix that is rectangular. All entries of $B, A$ are ...
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3answers
29 views

Find all the parameters and such that the line $y = ax + \frac{1}{2}a - 2$ intersects the hyperbola $xy = 1$ at right angles in at least one point .

Problem: Find all the parameters and such that the line $y = ax + \frac{1}{2}a - 2$ intersects the hyperbola $xy = 1$ at right angles in at least one point. My work: We try to find tangent to ...
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1answer
47 views

Can y' be squared? $\frac{dy}{dx} ye^{xy}=3x$

So the question I am trying to solve is: find $\frac{dy}{dx}$ if $ye^{xy}=3x$ I have tried: $\frac{d}{dx} ye^{xy}=\frac{d}{dx}3x$, factor out constant, and add y': $yy'\frac{d}{dx} e^{xy}=3$, ...
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2answers
29 views

Using implicit differentiation

I'm kind of stuck on this question. Use implicit differentiation to find the derivative of y =arccos(${\sqrt x}$) as a function of x and say where this derivative is defined. Don't really get the ...
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0answers
22 views

Implicit Differentiation Undefined Gradient

Find the equation of the tangent to the curve for the following equation at the point $(2, -3)$ $$4x^2-3xy-y^2=25$$ $$\therefore8x-3x\frac{dy}{dx}-3y-2y\frac{dy}{dx}=0 $$ $$\therefore\frac{dy}{dx}...
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1answer
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Determine the point on the curve $a ^ 2 x ^ 2 + y ^ 2 = a ^ 2$ in the first quadrant such that the area of ​the triangle by tangent

Problem: Let the $a$ arbitrary . Determine the point on the curve $a ^ 2 x ^ 2 + y ^ 2 = a ^ 2$ in the first quadrant such that the area of ​​the triangle by tangent the curve drawn at this point ...
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2answers
49 views

Prove that $F(x)=\alpha(x)f(x)$ is differentiable and compute the derivative

For an assignment, I have to solve this problem, but I just can't figure out how to continue. I already figured it out for the case $F(x)=f(x)+g(x)$ but this one I just cannot figure out. Would be ...
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1answer
39 views

Problem set derivation, tangent line and max/min of function

Find the numbers A,B such that the derivative function \begin{cases} Ax^3+Bx+2& \text{if }x=<2,\\ Bx^2-A & \text{if } x>2 \end{cases} is everywhere continuous . My work: Let's name the ...
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0answers
17 views

Separable DE Substitution

Hi I'm trying to solve a separable DE with a worked solution. Question: $y′ = (y−9x)^2$ Part of the Given Solution: Letting $v = y − 9x$ becomes $\frac{dv}{dx} = v(x)^2 − 9$ I get that $\frac{dv}{...
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3answers
69 views

Derivation and tangent problem set

Problem 1 On the curve $y=\frac{1}{1+x^2}$ find a point in which tangent line is parallel to the horizontal axis. My idea: Let's find $y'$. $$y'=\frac{-2x}{(1+x^2)^2}$$ If we want a tangent line to ...
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2answers
50 views

Find tangent to a curve that pass through the origin (implicit function)

I am trying to find the number of tangents to a curve that all pass through the origin. The curve's equation is $y=x^3+x^2−22x+20.$ I also need to find the equation of said tangents. My work: Let's ...