Questions tagged [implicit-differentiation]
For questions on finding and evaluating derivatives when a function is defined implicitly.
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Implicit differentiation of $\ln(x^2+xy+y^2)=1$ [closed]
Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ of $\ln(x^2+xy+y^2)=1$.
So for this natural log should I start by using the power rule or the the product rule of $xy$? The textbook talks about taking the ...
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Is the Primer Introduction to Derivation of Implicit Functions Using This Definition Ambiguous?
I'm exploring the concept of implicit function derivation and parameterized curve derivation, and I've encountered some points that seem vague to me. I would appreciate any insights or clarifications ...
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Find the derivative of $x^{\sqrt{y}}=y^{\sqrt{x}}$.
I am trying to solve:
$x^{\sqrt{y}}=y^{\sqrt{x}}$
Here's my solution. Please correct me if there is an error and kindly explain why.
$\sqrt y\ln{x}=\sqrt x\ln{y}$
$\left[y^\frac{1}{2}\ln{x}\right]^\...
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What exactly does $\frac{\mathrm{d}y}{\mathrm{d}x}$ calculate in implicit differentiation? (details given)
Does it calculate the derivative of the given equation?
Because in implicit differentiation we don't have $y= equation$ so I don't know what we are differentiating using the $\frac{\mathrm{d}y}{\...
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Implicit Differentiation: Proving a point is perpendicular at a certain point on two interesecting tangents.
Given the following question:
Consider the curves C1 and C2 defined as follows;
C1 : xy = 4 , x > 0
C2 : y^2 - x^2 = 2 , x > 0
Let P(a,b) be a unique point where the curves C1 and C2 intersect. ...
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How do I implicitly differentiate $\frac{d}{dx}(2x(y')^2y'')$
I know that I will probably need to use the 3-way chain rule:
$$(fgh)'=f'gh+fg'h+fgh',$$ but since I'm differentiating in terms of $x$, I'm very confused as to how to to "append" $y'$ terms ...
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Second implicit derivative
Given
$$x^2y+3y^3=7$$
I have to evaluate $y''$ at $(x,y)=(2,1)$.
I know
$$y'=-\frac{2xy}{x^2+9y^2}$$
hence, at $(x,y)=(2,1)$,
$$y'=-\frac{4}{13}$$
Therefore,
$$y''=-\left(\frac{(2xy)'(x^2+9y^2)-(2xy)(...
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Understanding Implicit Differentiation (intuitive)
Let's say we have:
$$ x^2 +y^2 = 25 $$
definition of differentiation:
$$\lim_{x\to h} = \frac{f(x+h)-f(x)}{h}=\frac{dy}{dx}$$
when we do implicit differentiation we have:
$$ df(x^2 +y^2) =df(25) $$?
...
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How to define implicit differentiation for non-functions?
How to define implicit differentiation for non-functions?
I want to refer to a similar question was asked here
am studying real analysis from the book “Introduction to Real Analysis” by Robert G. ...
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How do derivatives even work?
I'm a student currently looking into partial derivatives and was writing notes for a simple example.
$$\require{cancel}\text{find the derivative with respect to x:}\\x^2+xy-y^2=8$$ I got to the point ...
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Difference between the derivatives of arcsin(x) and arccos(x)
The derivative of $\arccos(x)$ and $\arcsin(x)$ are respectively $-\frac 1{\sqrt{1-x^2}}$ and $\frac 1{\sqrt{1-x^2}}$. However in the classic proof using implicit differentiation, when taking the ...
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Need help with Implicit differentiation for partial derivative
I am currently taking Cal 3 and my professor taught in class that if I have a function $F(x,y,z)=c$ (where $c$ is a constant), then $\frac{dz}{dx}=-\frac{F_x}{F_z}$. I understand the textbook ...
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Calculating the functional derivative of an implicit functional
We want to calculate the functional derivative of the following wrt $\rho$:
$$F=\int X dx$$
where $X$ is implicitly defined as:
$$X=\frac{1}{1+\overline \rho(x) \int \rho(x) X dx}$$
and $\overline \...
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implicit differential equation, y is a solution [closed]
Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution....
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Comparative Statics, implicit functions and second order derivatives.
I have that $x^{*}(w,z)$ and $y^{*}(w,z)$ is the implicit solution to a the system $F(x^{*}, y^{*},w) = 0$ and $H(x^{*}, y^{*},z) = 0$. Using the implicit function theorem, I can compute $\frac{\...
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Why do we call $y$ a function of $x$ in implicit differentiation?
When we have something like $y = 2x$ we understand $y$ to be the value of the function $f$ at each point $x$ where $f(x) = 2x$, to reiterate, $y$ is not a function but merely a label for the output of ...
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Mapping the derivative of an implicit function on a 2D plane
I'm in the middle of my first calculus class, a week ago we covered how to find the derivative of implicit functions, and I'm still thinking about it. I completely understand what I am supposed to ...
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Stuck on implicit differentiation exercise; inquiring about how to approach it
On exercise 8 on section 3.6 of Stewart’s Second Edition Single-Variable Calculus, it is asking me to find the derivative of y with respect to x by implicit differentiation. The problem it gives me is ...
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Multivariable implicit differentiation, $U=T\frac{\partial{S}}{\partial{T}}dT$
Given the equation $U=T\frac{\partial{S}}{\partial{T}}dT$ where $S$ is a function of V and T and V is being held constant, how do we then find $\frac{\partial U}{\partial T}$?
The way I'm explaining ...
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Is there an order to approaching implicit differentiation questions?
For example, I have the question "solve for $\frac {dy}{dx}$: $$ycos(xy)=y^2+2"$$ and I'm not sure if I should differentiate with the product rule or the chain rule first for the left side of the ...
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Use implicit differentiation to find dy/dx for $x \ge -2$. $y\sqrt{x+2}=xy+3$; I am getting a different answer than is given
$$y\sqrt{x+2}=xy+3$$ find $\frac {dy}{dx}$ for $x \ge -2$.
The given answer is $$-\frac{y(2(x+2)^{1/2}-1)}{2(x(x+2)^{1/2}-x-2)}$$
Here's my work so far:
$$y\sqrt{x+2}=xy+3$$
$$y\frac d{dx}\sqrt{x+2}+\...
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What happens when we take the derivative of an implicitly defined function?
Consider the function y(x) implicitly decided by this equation: $\sin(xy)=\cos(x+y)$.
The graph on plane looks like this:
According to my textbook, we can take the derivative of both sides with ...
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How do I find the horizontal tangents to the curve $x^3 + y^3 = 6xy$?
How do I find the horizontal tangents to the curve $x^3 + y^3 = 6xy$? James Stewart in section $3.6$ of the 7th edition (on page $167$) shows in a straightforward way that there's a horizontal tangent ...
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Chain rule (multivariate) and implicit differentiation problem
I have been worked out a solution for the following problem, but I am wondering if there is an easier way to solve it. I would be very grateful for any suggestions!
Let $z=f(x,y)$ a function of two ...
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Understanding an implicit solution to a nonlinear diffusion equation
I'm trying to understand an approximate implicit solution to a nonlinear diffusion equation from Perron (2011). The equation in question is
$$\frac{\partial z(x,t)}{\partial t} = \frac{\partial}{\...
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If $y=f(x)$ such that $(x+1)^3+(y+1)^3=16$ then evaluate : $\frac{d}{dx}\left(f(x+g(x)).g(x+f(x))\right)$ at $x=-1+\sqrt[3]{15}$
Let $f:\mathbb{R}\to\mathbb{R};y=f(x)$ be an explicit function defined by the implicit equation $$(x+1)^3+(y+1)^3=16$$ Let $g$ be the inverse of $f$.
Evaluate : $$\frac{d}{dx}\left(f(x+g(x)).g(x+f(x))\...
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Finding the area surrounded by a part of the implicit equation $\sin (y^x) = \cos (x^y)$
Finding the area surrounded by the part of the implicit equation $\sin (y^x) = \cos (x^y)$ such that $y\le 2n-x$ where $n$ is the solution to $n^n=\frac{\pi}{4}$ where $n<0.5$ bounded by the $x$ ...
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Computation of smooth Cholesky factorizations
Assume $X(t)$ is a time dependent positive definite symmetric matrix which satisfies the matrix differential equation
$\dot{X}(t) = Q(X(t)), X(0) = 0, $
where $Q(X(t))$ is a symmetric positive ...
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Solving logarithmic derivative in implicit form
I have the following function
$$
a\text{log}\left(X\right)+b\text{log}\left(y\right)=c\left[d\text{log}\left(X\right)+e\text{log}\left(y\right)\right]+\text{log}\left(\frac{A}{B}\right)
$$
I wish to ...
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How to solve an implicit differentiation while there are two expressions?
I have this problem:
if $$u=x^3y$$ then calculate $du/dt$ while $$x^5 + y =t$$ and $$x^2 + y^3 = t^2$$
I know that I should use the chain rule, and solved the problem to this point: $$\frac{du}{dt}=\...
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Two-variable Wronskian; regularity of coefficients
Let $f(x,y),g(x,y)$ be two real-analytic functions near a neighbourhood of $(0,0)$. Consider the following Wronskian determinant:
$$
D(x,y):=\det
\begin{bmatrix}
f(x,y) & g(x,y)\\
\partial_x f(x,y)...
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Implicit differentiation in terms of x, y, r and f
I just want to check am I practicing implicit differentiation correctly.
$$r = x+f(y+ rx + x^3)$$
where f is a differentiable function.
I am trying to use implicit differentiation to find $\frac{\...
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Calculus: Help with interpreting problem
I am currently working on implicit differentiation, and I am stuck with interpreting what specifically "in exactly one other point" means. Does this literally mean showing the derivative, or ...
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Show that $y^*(x)$ is increasing in $x$ if $\frac{\partial^2 f}{\partial x \partial y} > 0$.
Suppose $f : \mathbb{R}^2 \to \mathbb{R}$ is twice differentiable with non-zero second partials. If for every $x \in \mathbb{R}$, $\exists$ unique $y^*(x)$ that solves $\underset{y \in \mathbb{R}}{\...
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Trying to get region of stability of RADUA IIA
The method can be turned from its Butcher table into this set of explicit expressions:
$$k_1 = f\bigg(t_n + \frac{1}{3}h, y_n + \frac{5}{12}hk_1 + \frac{-1}{12}hk_2\bigg)$$
$$k_2 = f\bigg(t_n + h, y_n ...
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Butcher table to $y_{n+1}$?
I am trying to understand how to go from a Butcher table to something useful of the form $y_{n+1} = g(y_n, y_{n+1})$.
Let us first try the simplest case, which is backwards Euler.
Using the ...
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Algebraic manipulation before doing implicit differentiation
So if I have the curve $ y = x $ and I do implicit differentiation, I get $ y' = 1$
But when I do implicit differentiation of $ \frac{y}{x} = 1$ I get $y' = \frac{y}{x}$
I guess on all points on $ y = ...
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$xy=\ln (x+y)$ implies that $y=\frac{\ln x}{x}+O(\frac{\ln x}{x^3}), x\to+\infty$.
$xy=\ln (x+y)$ implies that $y=\frac{\ln x}{x}+O(\frac{\ln x}{x^3}), x\to+\infty$.
How to prove it? Even the easiest part $\lim_{x\to+\infty}y=0$ I have no idea.
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Find the answers of the two questions based on a rigorous function
Let $f$ be a differentiable function satisfying $$\sqrt[3]{f(x+y)}=\sqrt[3]{f(x)}+\sqrt[3]{f(y)}+1$$ $\forall \:\:x,y\in\mathbb{R}$ and $f'(0)=3$
If $h(x)=f(x)-x^3$ then find number of points where $...
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Sufficient conditions using implicit function theorem
Let $f:\mathbb{R^3} \to \mathbb{R}.$
Let $ (x_0,y_0,z_0)$ be a solution of the equation $f(x-y,y-z,z-x)=0$.
Find sufficient conditions such that extracting $z$ with respect to $x,y$ is possible.
I ...
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Some shortcut for system of equations with quadratic formulas
I have the following two-equation problem:
$$(A): \qquad x_2 \left( P_2 + 2 \frac{x_2}{v_2} \right) = x_1 \left( P_1 + 2 \frac{x_1}{v_1} \right) $$
$$ (B): \qquad M - \left( P_1 x_1 + \frac{x_1^2}{v_1}...
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Question about implicit differentiation in polar coordinates
I was wondering what was incorrect about this procedure that seems to lead to a contradiction. In polar coordinates its true that:
$x^2 + y^2 = r^2$
If I differentiate the equation implicitly with ...
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Question on Implicit Differentiation
I'm trying to solve this problem:
"Sand being emptied from a hopper at the rate of $10$ cubic feet per second forms a conical pile whose height is always twice its radius. At what rate is the ...
user1141413
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Coupled implicit partial differentiation
I'm struggling to determine two partial derivatives of the following two implicitly-defined, coupled equations.
Given:
$$x = X(x, y, E) = f_x(E) + g_x(x, y)$$
$$y = Y(x, y, E) = f_y(E) + g_y(x, y)$$
...
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Implicit Differentiation of $x^3+y^3=3xy$
Recently I came across a problem that asked to solve for the derivative of the following equation: $$x^3+y^3=3xy$$
I started by differentiating both sides, and I got $3x^2+3y^2\frac{dy}{dx}$ on the ...
user1141413
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why implicit partial differentiation and explicit partial differentiation of z with respect to x give different results?
I had a calculus exam and the question was to take the partial of z with respect ot x if: $$xe^y+yz+ze^x=1$$ I wrote z in terms of x and y: $$z= \frac{1-xe^y}{y+e^x}$$ I took the partial of the both ...
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What is really the TRUE definition of an implicit function?
First of all I would like to say that I have already found similar questions on stack exchange but somehow my confusion regarding the definition of an implicit function still linger.
The title says it ...
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Why can't I use the chain rule for multiple variables to differentiate the logarithm of a quotient
Say I want to find the derivative of the below function with respect to x, which equals zero.
$u(x,z) = \ln(\frac{\mathrm{x}^α + \mathrm{z}^{α}}{\mathrm{z}^α}),$$ $$z = h(x) = \mathrm{(a\mathrm{x}^4 + ...
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Can anyone explain implicit differentiation in an intuitive way?
I am in my final year of school. Understand the procedure and can do it well but do not understand why ? Can someone help ?
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How do I prove these two derivative expressions equivalent?
I was taking the implicit derivative of the following:
$$
\frac x y = x - y
$$
I'll cut straight to the chase and say my answer was this:
$$
\frac {dy}{dx} = \frac {y - y^2} {x - y^2}
$$
I've come ...
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