For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

learn more… | top users | synonyms

0
votes
0answers
8 views

Concept of inserting ansatz - separation of variables

In my textbook it says write the unknown function of two variables as a product of two functions of a single variable u(x, t) = X(x) T (t) but then the second step it goes straight away to have T ...
0
votes
0answers
35 views

How to find partial derivatives of this function? [duplicate]

Suppose $u = \sin \left(\left( x \sin^{-1} \left(y\right)\right)\right)$ Find $\partial u\over \partial x$, $\partial u\over \partial y$, $\partial ^2u\over \partial x^2$, $\partial ^2u \over ...
1
vote
1answer
23 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
0
votes
0answers
29 views

Proof of similar sign in second derivative

I have a function that is continuous, quasi convex, strictly increasing in $x$ and strictly decreasing in y. Furthermore it is always of the form $f(x-y)$, where f is arbitrary. How can I formally ...
-3
votes
0answers
21 views

Show that this function is differentiable at all points [on hold]

n-sphotos-h-a.akamaihd.net/hphotos-ak-xta1/v/t34.0-12/11116109_10206718706905332_835173146_n.jpg?oh=baf1ad15e0f70703e5eb93818b61c9d1&oe=55401033&gda=1430237746_5bfaae4f8271730ad293b579ab0e93ab ...
0
votes
3answers
25 views

Given that: $T(x,y)=\ \int_{x-y}^{x+y} \frac{\sin(t)}{t}dt\ $, calculate: $\frac{\partial T}{\partial x}(\frac{\pi}{2}, - \frac{\pi}{2})$.

Given that: $T(x,y)=\ \int_{x-y}^{x+y} \frac{\sin(t)}{t}dt\ $, How do I calculate: $\frac{\partial T}{\partial x}(\frac{\pi}{2}, - \frac{\pi}{2})$? I seriously have no direction for how to solve ...
0
votes
1answer
11 views

derivation of formula of the maximum likelihood

I found a derivation for the formula of logistic regression, it uses the hypothesis concept stated like: and the derivation that I have doubts is: For example, in the horizontal line that I ...
0
votes
0answers
22 views

What is $\frac{\partial}{\partial x}\int_0^t x(\tau)f(\tau)\, d\tau$? [on hold]

If $F(x,t)=\int_0^t x(\tau)f(\tau)\, d\tau$, What is $\frac{\partial}{\partial x}F(x,t)$ ? And what is $\frac{d}{dt}(\frac{\partial}{\partial x}F(x,t))$?
0
votes
1answer
15 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
0
votes
1answer
37 views

Derivatives - Show equality

Let $y(x)$ be defined implicitly by $G(x,y(x))=0$, where $G$ is a given two-variable function. Show that if $y(x)$ and $G$ are differentiable, then ...
0
votes
0answers
21 views

Finding this second order partial derivative

Want to find the second order of $$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial g}\frac{\partial g}{\partial x}+\frac{\partial z}{\partial h}\frac{\partial h}{\partial x}$$ Normally if I ...
1
vote
1answer
35 views

implicit multivariable derivative

I didn't really understand how implicit multi-variable functions are derived; I thought of another method which may fit and may not; suppose we have $xy^2z^3=8$ and we want to derive it; it is the ...
0
votes
0answers
27 views

Derivatives of semi-elasticities

I want to find the sign of $\partial_z (\partial_z \log F(T1) - \partial_z \log F(T2) )$ where $T1<T2$ and $F(T) = \int_0^T e^{g(t) + z\, h(t)} dt$ All we know is that $h(t)>0\forall t$ ...
1
vote
1answer
16 views

Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
1
vote
0answers
25 views

How to partially differentiate an integral with a density function?

I am given this result: $$\frac{\partial}{\partial x(t)} \left[\lambda \int u(x(t)) f(t) \mathrm{d}t\right] = \lambda u^\prime(x(t)) f(t)$$ Where $\lambda$ is a constant, and we have the probability ...
2
votes
1answer
49 views

Having trouble calculating $f_{xx}$ of a “variable-heavy” quotient.

Let $$ f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $$ Compute $f_x (0,0)$, $f_y (0,0)$, $f_{xx} (0,0)$, $f_{xy} (0,0)$, and ...
1
vote
1answer
21 views

Partial differential equations in $ \infty $

Suppose that we have $$\begin{cases} u_{tt}=u_{xx} \text{ for }\ 0 < x < \pi ,\ t > 0\\ u(x,0)=8\sin x \\ u_{t}(x,0)=0\\ u(0,t)= u(\pi,t)=0\end{cases}$$ find $ \lim_{t \rightarrow \infty ...
0
votes
1answer
46 views

What does it mean for partial derivative to be continuous and how does that imply differentiability?

In order for function to be differentiable at some point, it should be well approximated at that point. I understand that partial derivatives must exist, and that function needs to be continuous, but ...
2
votes
1answer
26 views

Smooth function conditions

A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
0
votes
2answers
43 views

Help Understanding Gradients

I understand that gradients are vectors with partial derivatives as components when working in 3D space, but does the the existence of a gradient at a point imply continuity at that point?
0
votes
2answers
18 views

Partial Derivative of Exponential Quotient

How would I go about finding the partial derivative with respect to $y$ of $z = (x^2/(1-y^3))^{0.5}$ The way I thought to do it was to get rid off the brackets and square root, making ...
0
votes
2answers
27 views

In differentiability

Let $f(x,y)$ and $g(x,y)$ are differentiable functions in $x$ and $y$. Suppose $f(x,y) = F(g(x,y))$.My question, Is $F$ differentiable function?!.
0
votes
1answer
26 views

How to calculate the partial derivative of an unknown function?

I have $$z= f(x-ay) + g(x+ay)$$ and I have to prove that $$\frac{\partial^2 z}{\partial y^2} = a^{2} \frac{\partial^2 z}{\partial x^2}$$ but I can't understand the way partial derivative works here. ...
0
votes
1answer
23 views

How to demonstrate this?

I've a question and it is: Evaluate ${\partial^2z \over \partial u^2}+{\partial^2z \over \partial v^2}$, if ${\partial^2z \over \partial x^2}+{\partial^2z \over \partial y^2}=0$ and $z=z(x,y)$, ...
0
votes
2answers
38 views

Help with Partial Derivatives and Chain rule

The question is Determine $f_{xy}$ when $f = ytan^{-1}(xy)$ I know that there is chain rule somewhere in here, but I don't understand where it comes from.
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
-3
votes
1answer
56 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} &= ...
0
votes
0answers
17 views

Jacobian of an error function containing state space vector (for linearization)

1) I have a vector of robot poses, that indicates the state space --> $X = \{(x_1,y_1),(x_2,y_2)....(x_N,y_N)\} \equiv \{a_1,a_2,...a_n\}$. 2) I have odometry measurement values at each time step ...
-1
votes
0answers
10 views

Partial derivation of three variables with respect to four variables

I have a set of equations relating variables $[K_1, K_2,K_3,K_4]$ to $[X, Y, Z]$ like: $$K_1=X-C Y cos(Z+ C_1)$$ $$K_2=X-C Y cos(Z+ C_2)$$ $$K_3=X-C Y cos(Z+ C_3)$$ $$K_4=X-C Y cos(Z+ C_4)$$ ...
1
vote
1answer
16 views

How to prove $\oint_{\partial D} \left( u \frac{d u}{d x} dx - u \frac{d u}{d y} dy \right) = \oint_{\partial D}u\frac{d u}{d \mathbf{n}} ds$?

For some simply connected region $D \subset \mathbb{R}^2$ with boundary $\partial D$, length differential along boundary $ds$ and normal $\mathbf{n}$, and some sufficiently smooth function $u(x,y)$ ...
0
votes
0answers
23 views

Increase of the rms during diffusion - Derivatives calculation

We consider a diffusion equation of the form $$ \frac{\partial F}{\partial t} = \frac{\partial }{\partial x} \!\cdot\! \left[ - \mathcal{F} (t , x) \right] \, $$ where ${ \mathcal{F} (t ,x) }$ is the ...
4
votes
2answers
49 views

Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 ...
1
vote
0answers
18 views

What does it mean to square a partial derivative with a one dimensional vector (scalar)?

Please bear with me.. In the above image, we need to substitute p2 from the partial derivative (pay attention to content in red boxes). If we're considering the one dimensional case, we can either ...
0
votes
0answers
29 views

Holomorphic functions (continuity of partial derivatives)

Let $f:\Omega\rightarrow \mathbb{C}$ be an holomorphic function i.e. for any $z_0\in \Omega$ there exists the limit: $$f^{'}(z_0) = \lim_{z\mapsto z_0}\frac{f(z)-f(z_0)}{z-z_0}.$$ Let us write $f(z) ...
1
vote
2answers
28 views

Straight vs Partial derivative

Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?
0
votes
0answers
33 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. ...
0
votes
1answer
37 views

Separation of variables for fourth order PDE

How do I solve: $$u_t = -u_{xxxx} + \pi^2u_{xx},$$ with BCs: $u_x(0,t)=u_x(1,t)=u_{xxx}(0,t)=u_{xxx}(1,t)=0$ and initial condition $u(x,0)=\cos(\pi x)$. We have been told that we can use ...
0
votes
0answers
6 views

Taking divergence of a field?

Given $J = (z-hut)J_osin[B(\frac {Δz}{2} - |z|)]$. I want to find $∇.J$, its confusing because I don't really see any r, theta, or psi directions. I only see z-direction, which isn't r because r is a ...
0
votes
1answer
21 views

Second partial derivatives (one seems strange)

I am trying to find the second-partial derivatives for the following equation: $$g = \sum_{i=1}^n \left(y_i - \frac{\theta_1 x_i}{x_i+\theta_2}\right)^2$$ Here, $\theta_1$ and $\theta_2$ are the ...
0
votes
2answers
27 views

What does this clause in my question mean for finding maximum/minimum values?

The question reads: Find the maximum and the minimum of the function $f(x,y) = 4xy$ subject to $x^2 + y^2 = 8$. I know to find the maximum and minimum, you must find the points where both $f_x ...
0
votes
0answers
18 views

Please check my calculations for finding a tangent plane to a parametric surface

Here's the question: Find the cartesian equation of the tangent plane to the surface $S : xy^2 + 3yz^2 − 2xyz = 1$ at the point $P(0, 3, 1/3)$. Here's what I did: The normal to the tangent plane at ...
3
votes
2answers
44 views

i'm looking for a proof of derivative of an inverse matrix

I'm trying to differentiate an inverse matrix in a form $ { \partial x^TA^{-1}x \over \partial A }$. I found the answer from the matrix cookbook as follows. $ { \partial x^TA^{-1}x \over \partial A ...
1
vote
1answer
27 views

partial differentiation and quotient rule

I want to compute a partial differentiation $\frac{\partial A}{\partial q}$ where A is ($\ddot{q}$), the output of standard manipulator equation, i.e. $$ H(q)\ddot q + C(q,\dot q)\dot q+G(q) =0 $$ ...
2
votes
0answers
47 views

Prove there exists a infinitely differentiable function whose value of partial derivatives of all orders at $0$ is a given function

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
2
votes
2answers
47 views

If $z^2 =f(x,y)$ then find $∂z/∂x$

If $z^2 =f(x,y)$ then find $∂z/∂x$ how to find $∂z/∂x$
2
votes
0answers
13 views

Please check my calculations involving directional derivatives and gradient vectors

Here is the question: If g is a function differentiable at $(a, b)$ such that $\nabla g(a, b) = (2, 3)$, then find all the vectors $\vec{u} = (x, y)$ with $x^2 + y^2 = 1 $ such that $D_\vec{u} g(a, ...
1
vote
0answers
17 views

Partial Differential Equations Coefficients

Given the differential equation $ A_{1}U_{xx}+A_{2}U_{yy}+A_{3}U_{xy}+A_{4}U_{x}+A_{5}U_{y}=U $ what does the value of $(A_{3})^{2}-A_{2}A_{1}$ mean whether its equal, less or larger than zero? I ...
0
votes
1answer
38 views

Notation for a derivative

I am interested if there is notation for a derivative that is in between a total derivative and partial derivative. The total derivative of $f(t,x,y)$ with respect to $t$ is $$ ...
0
votes
0answers
12 views

linear approximation of two variables function

Let $\displaystyle f(x,y)=\frac{\sin(x+y)}{\sin(x)}$. Find linear approximation of $f$ near $\displaystyle \left(\frac{\pi}{2},0\right)$. My try: The linear approximation of $f$ is ...
0
votes
0answers
47 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 ...