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

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4
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2answers
38 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 ...
1
vote
1answer
46 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 ...
1
vote
0answers
38 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 ...
0
votes
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\...
6
votes
4answers
111 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 ...
0
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2answers
32 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 ...
0
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0answers
14 views

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 ...
1
vote
1answer
33 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 ...
0
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0answers
28 views

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}...
1
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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(...
0
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0answers
23 views

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 ...
3
votes
2answers
89 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}+ (...
2
votes
1answer
31 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 ...
1
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3answers
52 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 ...
0
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3answers
27 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{...
1
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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} ...
1
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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 $...
3
votes
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} + ...
0
votes
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}{...
0
votes
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{...
2
votes
3answers
32 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 ...
1
vote
2answers
34 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 ...
3
votes
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 ...
1
vote
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 ...
0
votes
3answers
43 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 \...
2
votes
1answer
21 views

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 ...
2
votes
3answers
27 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 ...
1
vote
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': $yy'\frac{d}{dx} e^{xy}=3$, ...
1
vote
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 ...
0
votes
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}...
-1
votes
1answer
59 views

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 ...
1
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2answers
48 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 ...
0
votes
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 ...
0
votes
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}{...
1
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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 ...
0
votes
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 ...
1
vote
4answers
73 views

Tangent line problems

Problem 1 Find common tangent to the curve: $y+x^2=-4$ and $x^2+y^2=4$. My idea: Let $t1... y=ax+b$ is a tangent line to the first curve. Let $t2... y=cx+d$ is a tangent line to the second curve. But ...
0
votes
3answers
46 views

Prove to be hyperbolae $x^2 - y^2 = a$ and $xy = b$ intersect at right angle.

Prove to be hyperbolae $x^2 - y^2 = a$ and $xy = b$ intersect at right angle. My idea: $$h_1:=x^2 - y^2 = a$$ $$h_2:=xy = b$$ By using implicit differentiation we can find $h_1'$ and $h_2'$. $$x^2 - ...
1
vote
0answers
29 views

On the curve $xy^2 = 2a^3 , a > 0$ find all the points where the normal to the curve pass the origin.

On the curve $xy^2 = 2a^3 , a > 0$ find all the points where the normal to the curve pass the origin. My idea. Use formula for normal. $$(y-y_0)=-\frac{1}{y'(x_0)}(x-x_0)$$ So we need $y'$. We ...
0
votes
2answers
38 views

Find a tangent lines to the circle that tangent lines cross (4,1). $x^2+y^2+4x-2y-11=0$

Problem: Find a tangent lines to the cirlce that tangent lines cross (4,1). $$x^2+y^2+4x-2y-11=0$$ I try to find $y'$ and I get $\frac{-x-2}{y-1}$. And idea after that was using $y-y_0=\frac{-x_0-2}{...
1
vote
1answer
25 views

Implicit differentiation to determine area/angle of triangle

Let O denote the origin of the axis of co-ordinates, and let C denote the part of the parabola $y=x^2$ which lies in the first quadrant. A particle P starts at O and moves along C in such a way that ...
0
votes
1answer
43 views

Show that $y=e^{e^{cx}}$ is a solution of the differential equation $\frac{d^2y}{dx^2} =c^2 \cdot y \cdot \ln(y) (1+\ln(y))$

Question: Show that $y=e^{e^{cx}}$ is a solution of the differential equation $$\frac{d^2y}{dx^2} =c^2 \cdot y \cdot \ln(y) (1+\ln(y))$$ I know there are a lot of ways of solving this and I ...
1
vote
2answers
17 views

Regarding the derivative of an implicit line

Let $\sigma(x) = f(x,x^2+1)$, where $f: R^2 \rightarrow R^3$ is of class $C^1$ and $$Df(0,1) = \begin{bmatrix}0 & 1\\2 & 3\\4 & 5\end{bmatrix}$$ Find $\sigma ' (0)$. I know that the ...
1
vote
1answer
25 views

Need help finding the Horizontal Tangent of an Implicit Equation

I am given the equation: $(x^3) + (y^3) - (72xy) = 0 $. I found the derivative to be: $(-3x^2 + 72y)/(3y^2 - 72 x) $ I know that the numerator of the derivative must be set to $0$ in order to ...
1
vote
1answer
60 views

Help with implicit differentiation problem [duplicate]

Here is the problem: A ladder 15 metres high is propped up against a high wall. The bottom of the ladder slides away from the wall at a rate of $1\ {m/s}$. How fast is the top of the ladder ...
1
vote
1answer
48 views

Error in differentiation/integration problem

(Edited to directly present the problem only, for any future readers. The original question can be read in the revision history.) Is the following correct? $$\frac{d}{dy}\left(\int y\,dx\right) = x$$...
0
votes
1answer
9 views

How to tell the monotonicity from the implicit expression

I have an equation as $f(x,y)=0$, and $f$ is very complicated that there is no way to rewrite it as $y=g(x)$ explicitly. I would like to show that $y$ is increasing at $x$, is there any way I can ...
0
votes
0answers
32 views

Implicit Function Theorem system problem

For the system of equations, first check if implicit function theorem can be satisfied at $A=(x,y,t)=(1,1,0)$, $x$ and $y$ are variables and $t$ is a parameter. Then find the expression for $\partial ...
1
vote
1answer
48 views

Why can't I change an equation before I differentiate it?

So recently I was reviewing calculus, and I tried to differentiate the equation: $(x^2-y^2)/(x^2+y^2)=1/2$ The first thing I did was make the equation easier to differentiate by multiplying the whole ...