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

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3
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2answers
30 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
29 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
40 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
20 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
26 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
46 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
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2answers
27 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
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0answers
21 views

Implicit Differentiation Undefined Gradient

Find the equation of the tangent to the curve for the following equation at the point $(2, -3)$ $$4x^2-3xy-y^2=25$$ $$\therefore8x-3x\frac{dy}{dx}-3y-2y\frac{dy}{dx}=0 $$ ...
-1
votes
1answer
56 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
45 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
37 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 ...
1
vote
3answers
67 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
47 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
70 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
45 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
30 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 ...
1
vote
1answer
24 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
0answers
13 views

Graphically interpretation of a function and its derivative?

I know that a derivative of a function gives you the tangent to the original function at a given point, but is there a relation between the graph of the function and its derivative? Sorry for asking ...
0
votes
1answer
41 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 ...
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
53 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
47 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) = ...
0
votes
1answer
8 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
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0answers
30 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
47 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 ...
1
vote
1answer
54 views

For the differentiation of $x^{\frac23} + y^{\frac23} = a^{\frac23}$, why is the substitution $x = a \cos^3\theta$ legal?

While looking at a solution for finding the derivative of $x^{\frac23} + y^{\frac23} = a^{\frac23}$, the book uses: Let $x = a \cos^3\theta$ and $y = a\sin^3\theta$ However, why would that ...
0
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0answers
21 views

Gauss legendre stability region

I was reading this part, the book is originally from the following link, page 296 ...
0
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2answers
27 views

Solve for $\frac{dy}{dx}$ of a trigonometric function after implicit differentiation

I'm supposed to implicitly differentiate $\sin(x+y)=2x-2y$. I've already taken the first derivative and got $$ \left(\frac{dy}{dx}+1\right)\cdot\cos(y+x)=-2\left(\frac{dy}{dx}-1\right) $$ ...
0
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0answers
24 views

Related Rates baseball problem. Can anyone explain this to me with a good picture or something like that

A baseball diamond is 90 feet by 90 feet and the pitcher's mound is at the center of the square (pretend it is). If a pitcher throws a baseball at 90 miles per hour , how fast is the distance between ...
0
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0answers
23 views

Show that Runge Kutta Gauss Legendre is 4th order

So I have this Butcher array of \begin{array}{c|ccc} \frac{1}{2}-\frac{\sqrt{3}}{6} & \frac{1}{4} & \frac{1}{4}-\frac{\sqrt{3}}{6}\\ \frac{1}{2}+\frac{\sqrt{3}}{6} & ...
1
vote
1answer
58 views

Implicit Function Theorem Application to 2 Equations

So I think I understand how to use the Implicit Function theorem to find partial derivatives given one function but I am confused as to how to do this for 2 functions. I'm trying to find ...
1
vote
0answers
19 views

Second order 1-D wave equation, Implicit scheme

I am trying to solve the second order wave equation in 1 dimension from the implicit method by finite difference. $$ \frac{\partial^2U}{\partial x^2} = \frac{\partial^2U}{\partial t^2} , \ t>0 ...
3
votes
2answers
86 views

Theorem about implicit solutions of differential equations?

I have a theorem in my book (in the context of separation of variables method) which I don't fully understand: Suppose $g=g(x)$ is continuous on $(a, b)$ and $h=h(y)$ is continuous on $(c, d)$. ...
2
votes
1answer
47 views

Derivative of $(log (x))^{x}$

How can we calculate the value of $ \frac{dy}{dx} (log (x))^{x}$ I tried doing it the following way : Let $ y= (\log (x))^{x} $ $ \log y = x \log \log (x)$ Then differentiating both sides with ...
0
votes
1answer
35 views

Tangent line problem with implicit differentiation

Given: $[\tan^{-1}(x)]^2+[\cot^{-1}(y)]^2=1$ Find the tangent line equation to the graph at the point $(1,0)$ by implicit differentiation I found the derivative: ...
0
votes
1answer
29 views

Finding Differential in Implicit Function theorem questions

Let $\Phi:\Bbb{R}^2\to \Bbb{R},w=\Phi(u,v)$ a $C^1$ map and let $a,b\in \Bbb{R}$ such that $\Phi(0,0)=0$ and $a\Phi_u(0,0)+b\Phi_v(0,0)\ne 0$. Show there exists a neighborhood of $(0,0,0)\in ...
1
vote
0answers
19 views

Euler Schemes in Stochastic Differential Equations

So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations I ll start with explicit. Say i have the following SDE known as ...
1
vote
1answer
31 views

Implicit differentiation which respect to which variable?

The height of a cone is 6 inches, and, at it's opening it has a 3 inch diameter. Find a formula for the height as a function of the volume. Compute $h'(V)$ So if the volume, $V$ of a cone is $V = \pi ...
2
votes
0answers
43 views

Getting tangent to point on a function in 2 variables

We have a function $h(x, y) = \ln(x^2 - y) + x^2y + 4 \cos(\pi(y - x))$ in two variables. First we want to show that $h(x, y) = 8$ can be solved for $y$ in point $(2, 3)$. This is an implicit ...
9
votes
1answer
131 views

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

Consider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function ...
0
votes
0answers
23 views

Implicit - simplify last step

Please let me know if this link works. I'm pretty new at posting questions. Maybe there is a better way to post from the derivative-calculator.net site. I'm not sure how they simplify the last step ...
0
votes
2answers
27 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
1
vote
1answer
26 views

Generalize Implicit Differentiation to find Tangent Plane

For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ ...
6
votes
1answer
113 views

What's the arc length of an implicit function?

While an explicit function $y(x)$'s arc length $s$ is easily obtained as $$s = \int \sqrt{1+|y'(x)|^2}\,dx,$$ is there any formula for implicit functions given by $f(x,y) = 0$? One can use the ...
0
votes
0answers
43 views

Ordinary differential equations as variational problems

Considering an ordinary differential equation of first order in the implicit form $$ F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0 $$ with $\alpha,\, q_0$ constants, what is the relation of the solution ...
0
votes
2answers
35 views

Find the derivative $y'_x$ from the equation $y^3 + x^2 = xe^{y^2} - y\sin x$

Find the derivative $y'_x$ (edit: $y'_x = {dy \over dx}$) from the equation $$y^3 + x^2 = xe^{y^2} - y\sin x$$ I generally don't understand how it should be done. Should I just implicitly ...
-1
votes
1answer
64 views

Calculation of Rise in Height of water in a Frustum of Right Circular Cone

A volume of frustum of right circular cone is calculated as follows. With known h, R & r of a container with the shape shown below, how to find out the rise in height for each time $7m^3$ of water ...