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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.

If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by: $f(x,y)= \begin{cases} (x,y-x^{2}) & if & x^{2} \leq y \\ (x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
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2answers
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Finding the position $s(t) $ of a particle that has acceleration defined as $a(t)=3t+5$.

I am a Calculus 1 student, and we're learning about antiderivatives. I've run into a problem I'm not sure how to solve. A particle moves with acceleration defined by $a(t) = 3t+5$. Find the ...
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3answers
42 views

How to find the general antiderivative of $f(x)=x(6-x)^2$?

I want to find the general antiderivative of $f(x)=x(6-x)^2$. However, I keep getting it wrong. I am new to antiderivatives, but I think the first thing I should do is differentiate. $$\frac{d}{dx}(...
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1answer
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Why do we take magnitude into account when calculating the directional derivative?

Given that the directional derivative is defined formally as: $$ \nabla_\vec{v}\, f\left(\vec{x}\right) = \lim_{h \to 0} \frac{f\left(\vec{x} + h\vec{v}\right) - f\left(\vec{x}\right)}{h|\vec{v}|} $$ ...
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42 views

Functions that have the same derivative

Let’s say I have two continuous functions $f(x)$ and $g(x)$ , and both have the same derivative $h(x)$. How could I formally show that $f(x)=g(x)+c$ where $c$ is a constant. I know I have to show that ...
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1answer
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Strategies for proving continuity and differentiability of trigonometric series

Let $f$ be a function defined by a series $$f\left(x\right)=\sum c_n e^{inx}.$$ Sometimes, I can prove that the series converges pointwise (when it does), using the Dirichlet test. When the ...
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Derivative of integral for non-negative part functions

Suppose we have $N$ random variables $u_1,u_2,...,u_N$, which are i.i.d with PDF $f(\cdot)$. Then how to compute the partial derivative of $g(x,y)$ with respect to $x$ and $y$? That is, $\frac{\...
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2answers
42 views

Generating function of binomial coefficients

We want to evaluate the sum $$\sum_{L=0}^{\infty}\frac{1}{2}L(L+1)x^L$$ From this set of notes (page 2, equation 8) we find the formula $$\sum_{n=0}^{\infty}\binom{n}{k}y^n = \frac{y^n}{(1-y)^{n+1}}$$...
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Norml, Application of derivatives

If $x+4y =14$ is normal to the curve $y^2=αx^3 - β$ at $(2,3)$, then the value of $α+β$ is? I equated the slope of the normal with the value of $-dx/dy$ and found $α=2$, how do I find $β$?
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1answer
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For continuously differentiable $f,$ is it true that the set $\{(x_0,x_1)\in (0,1)^2: |f(x_0)| + |f'(x_1)| \geq \epsilon\}$ not compact in $(0,1)^2?$

Notations: We denote $C_0^1(0,1)$ the collection of all real-valued continuously differentiable function $f$ on $(0,1)$ that vanish at boundary, that is, for any $\epsilon>0,$ the set $$\{x\in (0,...
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1answer
38 views

How does $\frac{-e^{-x}}{\sqrt{e^{-2x}-1}}$ rearrange?

I have been revising for my engineering mathematics exam which has a multiple choice question in it, which asks the following: The derivative of $\arcsin(e^{-x})$ equals: With several possible ...
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1answer
50 views

When do integrable functions have a primitive?

Studying hyperbolic partial differential equations, we arrive, in a certain calculation, to the following doubt: every integrable function has a primitive? If $ u_0 $ is integrable, then $ \exists ...
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3answers
23 views

Differentiating a complex function using the definition

I need to differentiate the complex function $f(z)=z^2+z$. I know that the definition of a derivative is $f'(z)=\frac{f(z)-f(z_0)}{z-z_0}$. In this case, $f'(z)=\frac{(z^2+z)-(z_0^2+z_0)}{z-z_0}$. ...
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3answers
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Proof of derivative of $x^TBx$ using the product rule

I'm trying to prove that when $f(x) =x^TBx$, then $f'(x) = (B + B^T)x$. I haven't found this formula online but going through the calculations using index notation this is what I came up with. This ...
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2answers
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Non-Separable Polynomials and their Derivatives

We say that a polynomial $f(x) ∈ F[x]$ is inseparable if it has a repeated root in some field extension. Otherwise we say that $f(x)$ is separable. Prove that $f(x)$ is separable $\iff\gcd(f, Df) = 1$....
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Need help with this differential equation $\frac{dh}{dt}=\frac{5}{h^2}-\frac{1}{20}$

I am an A level student and am stuck on this differential equation. I know it is simple for most of the brilliant minds here but I have been trying for an hour with no good result. $$\frac{dh}{dt}=\...
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Why we sum up the derivatives of the loss w.r.t. Weights at each time step in RNN back-propagation?

I am reading a paper explaining the derivations of the back-propagation equations in RNNs. There I read 'Note that the Weight Matrix remains the same across all time sequence so we can differentiate ...
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Is $f\left( x \right) - cx$ unimodal when $f$ is strictly increasing and log-concave?

Let $f$ be a twice differentiable, strictly increasing and strictly log-concave function. Let $c$ be a positive constant. Is $f\left( x \right) - cx$ unimodal? More precisely, I would like to know ...
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2answers
48 views

Smooth approximation of $f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$ [on hold]

I'd like to find a smooth function to approximate $$f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$$ This function should be differentiable everywhere. Thanks.
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1answer
37 views

Derivative with a sum

I am going crazy over a relatively simple question. I want to find the derivative with respect to $\theta$ of $\frac{1}{\theta} \sum_{i=1}^{n}(y_{i} - \text{log} \theta)$ Using the product rule I ...
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30 views

What intrinsic property determines whether a function is analytic

Given we know the value of all order derivatives at a point $x_0$ for a given f(x). As per my knowledge all the geometric properties like slope, curvature, convexity are functions of solely the ...
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1answer
19 views

Derivative of polygamma function

I am working on my Matlab homework and I have to make a derivative of function $f(x)=\psi (x)\cdot \sin (x)$ , where $\psi(x)$ is polygamma function. What the derivative of $\psi(x)$ will be?
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3answers
22 views

Take Derivative Of Logarithm with Coefficient

IN my books, I see this example that derivative of $ln(\frac{1}{x^3})$ is $\frac{-3}{x}$. I know derivative of just $ln x$ is $\frac{1}{x}$. What is rule for multiplying coefficient with 1/x? I not ...
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Is there a solution to $(c+x)f'(c+x)=c(x-f(x))$?

(possibly under the condition f(0)=0). In general, is there a name for differential equations which feature the function f evaluated at another function of the argument, e.g. something like f(g(x))=f'...
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1answer
52 views

Not Sure Why Limit Is In Book

I am looking at logarithms and derivatives. In my books, Bostock and Chandler, it saying: $\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0}(\frac{1}{\frac{\...
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1answer
33 views

How to take the derivative of $x_{ab} = \sum_{i=1}^{c} A_{ia}B_{ib}$

I have an equation of $$x_{ab} = \sum_{i=1}^{c} A_{ia}B_{ib}$$ Where $A \in \mathbb R^{c \times a}, B \in \mathbb R^{c \times b}$. How to compute the derivative ? $$\frac{\partial x_{ab}}{\partial ...
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Derivative of function $y=\int f(x) dx$ [on hold]

Let $f(x)$ be a continuous function on $[1, 3]$. What is $\frac{\mathrm dy}{\mathrm dx}$ if $y = \int f(x)\mathrm dx$ ?
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Doubt in the statement of $n$th Derivative Test.

My definition of local maximum : $x=c$ is the point of local maximum, if $\exists h>0$ such that $\forall x \in (c-h,c+h) \Rightarrow f(c) ≥ f(x)$ Similar definition for local minimum, and ...
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Derivative of $C = D_KL(p || q)$ w.r.t $q_{ij}$ where $q_{ij} = \frac{\exp(z_{ij})}{\sum_{k=1}^N\sum_{l=1}^N \exp(z_{kl})}$

For an exercise, I need to compute $$\frac{\partial C}{\partial q_{ij}}$$ where $$C = D_{KL}(p || q) = \sum_{i=1}^N \sum_{j=1}^N p_{ij} \log \left (\frac{p_{ij}}{q_{ij}} \right)$$ and $q_{ij} = \...
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Help understanding proof of Euler's Homogeneous function theorem when t=1

In the proof of Euler's homogeneous function theorem here The start with the definition of a homogeneous function $$ f(tx,ty)=t^n f(x,y) $$ and take the derivative of both sides w.r.t $t$ which gives ...
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Prove $\exists\theta\in(0,1)$ s.t. $\Delta f=\frac{\partial f}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y$

Let $f(x,y)\in C^1$ in $\mathbb{R^2}$ and let $(x_0+\Delta x,y_0+\Delta y)$ and $(x_0,y_0)$ be points in $\mathbb{R^2}$. Prove that $\exists\theta\in(0,1)$ such that: $$f(x_0+\Delta x,y_0+\Delta y)-...
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2answers
34 views

Check whether $F(x,y)$ is differentiable or not at $(0,0)$

So...I am having trouble checking whether $F(x,y)=e^{x+y}$ is differentiable or not on $(0,0)$ The partial derivatives for $x$ and $y$ on $(0,0)$ (if no mistakes were made) are $1$ for both of them. ...
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0answers
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Generalization of Rolle's Theorem

It is known that if a differentiable function f has two equal values f(a)=f(b) , then there is a value f(c) between them which is a maximum or minimum for that interval. However, if the one sided ...
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Basic differentiation problem using multiple rules

$$\text{Find}~ \frac{\,dy}{\,dx}~, \qquad \text{where}~~y =\frac{\sqrt{2x^2}}{\cos x}$$ Here's the basic question. The solutions suggest to use the quotient rule for the top half and bottom half and ...
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1answer
30 views

Help with implicit derivation

Two straight paths from a point P make an angle of 60 deg. Astrid is jogging on one of the paths, while Beth is jogging on the other. At a point in time: Astrid is at 50m with a speed 3.1m/s. Beth ...
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1answer
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If a multivariate function has a gradient at $x$, then it is always differentiable at $x$, right?

By being differentiable, I mean a function of several real variables $f: \mathbb{R^m}\rightarrow \mathbb{R}$ is said to be differentiable at a point $x_0$ if there exists a linear map $J: \mathbb{R}^...
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Is it true $\int_0^k \frac{\partial f(x,y)} {\partial x}\,dx = f(k,y) + \textrm{a function of }y$?

I saw it written somewhere that $$\int_0^k \frac{\partial f(x,y)} {\partial x}\,dx =f(k,y) + \textrm{a function of }y$$ This seems feasible, but I haven't seen the integral of a partial derivative ...
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1answer
9 views

Derivate of inverse of composite function

I'm very confused, and this is probably a stupid question. I want to calculate $ \frac{d}{dx} f^{-1}(g^{-1}(x))$. However, I get two seemingly different results taking two different approaches. I. $\...
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1answer
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A differentiation question on differences between calculating it through 2 ways

What is result of differentiation of $y=\sqrt{ 25-x^2}$ and why it’s different when calculating differential of implicit function $x^2+y^2=25$?
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Calculating many partial derivatives of an infinitely differentiable function.

Let $f(x)\in C^{\infty}(\mathbb{R}^{n})$, $b$ real number. Compute $\partial_{x_{i_1},\ldots,x_{i_k}}f(x)^{b}$ with $1\leq i_j\leq n$ all diferent How can I find a simple formula to calculate the ...
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1answer
32 views

Partial derivative of $f(u,v)$

Let $f(u,v) = c$ where $u(x,y) , v(x,y)$ are functions and $c$ is constant. Can we conclude $\frac{\partial f}{\partial v} = \frac{\partial f}{\partial u} = 0$ ? It really sounds confusing to me but I'...
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Total and Partial Derivatives With Multiple Implicit Independent Variables [on hold]

Given a function of function(s) of multiple independent variables, i.e., $f(x(s,\ t))$ or $f(x(s,\ t),\ y(s,\ t))$, does $f$ have a total derivative? Does $f$ have partial derivatives with respect to ...
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30 views

Derivative of depended vector/matrix by matrix multiplication

If $X \in R^{N\times N}$ and $f(X): R^{N\times N} \rightarrow R^N$ How can I compute the derivative (where $\times$ is a vector by matrix multiplication) $\frac{\partial Y(X)}{\partial X} = f(X) \...
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33 views

How to get inequality ‎$‎0‎\leq ‎\gamma‎_g + ‎log ‎g(1)\leq ‎\frac{g^‎\prime_{-}(1) + g^‎\prime_{+}(1) ‎}{2g(1)}‎$‎?

‎Let ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎be a‎ ‎real‎ ‎function ‎such ‎that ‎‎$\log g(x)$ ‎is ‎concave and ‎$‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ ‎for each ‎$‎w&...
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26 views

Differentiate under expectation sign.

Let $f(x)$ be the probability density function of a $\text{Uniform}(0,1)$ distribution. Let $Z\sim\text{Normal}(0,1)$ and $g(x)=E[f(x+Z)]$. If one plots $g(x)$ in a computer, one observes like an ...
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1answer
23 views

Directional derivative and Jacobian matrix

I have a problem with an exercise that goes as follows: Let $\mathbf{f}$ be a $\mathbb{R}^n\rightarrow \mathbb{R}^m$ function and $\mathbf{a}$ an interior point of the domain of $\mathbf{f}$. ...
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2answers
34 views

Numerical differentiation order

For the derivative approximation $ f′(x) $ of a function $ f(x) $ below: $$ f′(x) ≈ \frac{4f(x + ∆x) − 3f(x) − f(x − 2∆x))}{6∆x} $$ a) What is the approximation order of the formula? b) Why is it a ...
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1answer
168 views

What does $F'$ and $F''$ mean?

I'm trying to learn what a Taylor series is, This is the equation I'm looking at and I know 0 calculus. I have been told that $F'(x)$ is a derivative but what does $F''(x)$ mean?
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1answer
31 views

Proving some derivative properties

I want to prove the following for $f:[a,b] \to \mathbb{R}$. (1) If $f$ differentiable in $(a,b)$, and $\lim_{x \to a+} f'(x)$ exists, then $f'(a)$ exists. (2) If $f'(z)$ exists for $z \in (a,b)$ and ...
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1answer
39 views

A differentiable multivariable function

I know the chain rule for the multivariable functions works when the partial derivatives are continous but if the function is just differentiable does the chain rule work ? I mean if $z = f(x,y)$ is ...