Questions tagged [partial-derivative]
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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.
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Show the partial derivative and continuity of the function below.
The function is defined to be $$ f(x,y) = \begin{cases} (x^2+y^2)\sin(\frac{1}{\sqrt{x^2+y^2}}) , & \text{$(x,y)\neq(0,0)$} \\ 0, & \text{$(x,y)=(0,0)$} \end{cases}. $$ Prove that the partial ... continuity partial-derivative- 59
Simplifying equations of partial derivatives by substitution
Let $u=\ln x$, $v = \ln y$. Then what is another form of $$x^2 \left( \frac{\partial^2 f}{\partial x^2}\right) + y^2 \left(\frac{\partial^2 f}{\partial y^2}\right) + x \left( \frac{\partial f}{\... calculus derivatives partial-differential-equations partial-derivative substitution- 139
${f}\begin{pmatrix}x \\y \\\end{pmatrix}$=$\begin{pmatrix}x^2-y^2 \\2xy \end{pmatrix}$ is differentiable at each point
I need help with understanding some steps of a task. I tried to solve the uncertainty by myself by using "Approach zero" but the problem is, that I am not able to write these kind of ... differential-topology partial-derivative- 292
Deriving SDE of Forward Rate
We have $$ F(t,T,+\delta)=F(0,T,T+\delta)e^{\int_{0}^{t}\beta(u,T)dW_{u}+0.5\int_{0}^{t}\beta^{2}(u,T)du} $$ How would we derive the SDE of the above equation? Taking partial derivatives, I got: $$ \... stochastic-processes partial-derivative stochastic-differential-equations- 1
Product notation in partial differentiation
For a function $f:\mathbb{R}^n\to \mathbb{R}$, is it correct to write, for any $n\in \mathbb{N}$, the expression $$ \frac{\partial^n f}{\partial x_1\cdots \partial x_n}=\frac{\partial^n f}{\prod_{i=1}^... functions derivatives notation partial-derivative products- 3,418
How can I prove this partial derivative question? [closed]
Let $z=y\ln(x^2-y^2)$. Prove that $\dfrac1x \dfrac{\partial z}{\partial x}+ \dfrac1y \dfrac{\partial z}{\partial y} = \dfrac{z}{y^2}$. calculus derivatives partial-derivative- 9
Swapping partial derivatives and limits
What conditions does a function have to meet for the following to be true? $$\displaystyle\lim_{x\to 0}\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}\lim_{x\to 0}f(x,y)$$ I've looked ... limits multivariable-calculus partial-derivative- 1
What is my error in this $\nabla_{\vec{v}} f(x,y,z)$ at $\vec{a} = (-1, -1, 4)$ and $\vec{v} = (\frac{\sqrt 2}{2}, \frac{1}{2}, \frac{1}{2})$ problem
I want to find gradient of $f(x,y,z) = \sqrt{xyz}$ in the direction of $\vec{v}$ at a point $\vec{a}$. That is, $\nabla_{\vec{v}} f(x,y,z)$ at $\vec{a} = (-1, -1, 4)$ and $\vec{v} = (\frac{\sqrt 2}{2},... multivariable-calculus partial-derivative vector-analysis- 33
Is the following vector expression true?
Suppose I have the following equation involving two vectors $\vec{A}$ and $\vec{B}$ that satisfy the following relation : $$\vec{\nabla}\times\frac{d\vec{A}}{dt}=k\frac{\partial \vec{B}}{\partial t}$$ ... multivariable-calculus vectors partial-derivative vector-analysis curl- 415
What is the partial derivative of a section?
I made an attempt to understand the jet bundle after reading up on fiber bundles but I got stuck at the first definition, how is the partial derivative of a section defined? The total space contains ... partial-derivative jet-bundles- 637
Derivatives are not matching textbook. Are my assumptions or calculations wrong?
I am trying to follow a derivation in a book, but I get a different result and can't figure out what I'm missing. I have a system of equations like this: \begin{align} \dot{r} &= v \\ 0 &= g(r)... calculus derivatives partial-derivative- 147
What is $\nabla f$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ (real to real)
Let's say $f$ is a function that converts a real number to a different real number. In other words, $f: \mathbb{R} \rightarrow \mathbb{R}$. What is the gradient $\nabla f$? In this case, is $\nabla f(... linear-algebra elementary-number-theory partial-derivative gradient-descent- 121
Trying to understand a solution to a problem regarding chain rule and partial derivatives
I'm trying to understand the provided solution yet something doesnt seem right, I'll share the problem, my solution (Which is incorrect according to my lecturer), and the lecturer's solution. The ... partial-derivative chain-rule- 163
How to find partial derivative function $ P_i = \frac{b_iz_i^k}{z_i^k+\theta_i^k} $
I have a function that converts a vector of real-valued variables $\vec z$ to a probability $\vec P$. In the elementwise version of this function, $i = 1, 2, ..., K$ and $b_i = \frac{z_i}{\sum_{v=1}^... calculus probability vectors partial-derivative vector-analysis- 37
How a gradient gives direction of steepest ascent in gradient decent algorithm to find direction of minima
I have read about Directional derivatives and know that to maximize the dot product of gradient vector and u(vector) we have to make cosine 0 deg as its 1. But after we do this we are only left with ... vector-spaces partial-derivative machine-learning gradient-descent neural-networks- 11
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