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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4 views

Is the following notation correct for $(af)'=af'$?

I want to show that $(af)'=af'$ where $a$ is a constant, i.e. $d(a)=0$. Is the following approach correct? I prefer Leibniz notation. I don't think that $d(af)=\lim_{h \to 0}{\frac{\bigl(a*f(x+h)\bigr)...
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23 views

Possible pattern in higher (mixed) partial derivatives of the function $f(x,y)=\exp(xy)$

I was writing down the $4^{\mathrm{th}}$ degree Taylor polynomial of the function $f:\Bbb R^2\to\Bbb R^+, f(x,y)=e^{xy}$ around $(0,0)$ so I computed all the necessary partial derivatives: $$\begin{...
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7 views

derivative of quadratic form with respect to orthogonal matrix

I have the quadratic form: $$ f(P;Y,\Lambda) = Y^\top P\Lambda P^\top Y, $$ where $\Lambda$ is a diagonal $p\times p$ matrix of real (positive) values, $P$ is an real orthogonal matrix, and $Y$ is a $...
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43 views

$\displaystyle\exists\lim_{x\to\infty}f(x)\Rightarrow \lim_{x\to\infty} f'(x)=0$ [duplicate]

I know that if $f$ is continuously differentiable and $f'$ is uniformly continuous then we have $$ \exists\lim_{x\to\infty}f(x)\Rightarrow \lim_{x\to\infty} f'(x)=0. $$ But I was thinking of changing ...
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31 views

Determining whether the evaluation mapping $p \mapsto \frac{d}{dx}\bigg|_p$ is smooth

For some reason, this quetion gives me a mental block: Suppose that we have the mapping $f(a, p) = a\cdot \frac{d}{dx}\bigg|_p$ for $a, p \in \mathbb{R}$, and we'd like to argue why this is a smooth ...
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13 views

Partial Derivative of Maximization function

I am trying to calculate a partial derivative of a utility function that itself is a maximization, where c is a constant parameter, x the decision parameter that is chosen to maximize the utility/...
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Find derivative of inverse of function $y=2x^3-6x$ and calculate it's value at $x=-2$.

Find derivative of inverse of function $y=2x^3-6x$ and calculate it's value at $x=-2$. My Approach: We know that $(f^{-1}(f(x)))=x$ Taking derivative both side $(f^{-1}(f(x)))' \cdot f'(x)=1$ $(f^{-1}...
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Linear Second Order differential equation

I am working through a derivation of a second order differential equation from this linked document. I have a question about the second derivative of (Y + y(t)) and getting y"(t). Is it because ...
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1answer
30 views

Is it possible to use the L'hopital rule for computing the derivative of this function at x=0?

The function \begin{equation} f(x)=\begin{cases} \cos(1/x) & x\neq 0\\ 0 & x=0\end{cases}. \end{equation} and the function $F(x)=\displaystyle \int ^x_0 f $ And I know that by considering ...
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Suppose that $\ln(f(t))= t f(t)$. What is $\lim f'(t)$ as $t \to 0^+$?

Suppose that $\ln(f(t))= t × f(t)$, what is $\lim f'(t)$ as $t$ goes to $0$ from the positive side? Using implicit derivative, I got that $f'(t)=\frac{f(t)^2}{1-tf(t)}$. But i got a bit confused while ...
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2answers
44 views

Applying chain rule to $(e^x)^n$ [closed]

When applying chain rule to the $n$-th power of a function of $x$, the power of the function is reduced to $n-1$. Why doesn’t this occur when taking the derivative of $(e^x)^n$?
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Finding non-zero function such that $f(2x)=f'(x)\cdot f''(x)$

I am thankful if someone can help me or show me the clue. As honestly as possible, I got stuck on this problem. I need some help in finding a $f(x)\neq 0$ such that $$f(2x)=f'(x) \cdot f''(x),$$ where ...
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Is the log in the simplified gradient equation a $\log_{10}$ or a $\log_e$?

I am trying to understand how does this youtube video simplifies the gradient equation ($\nabla f(x) + u\nabla g(x)=0)$ with the KKT conditions. Indeed, according to the Youtuber: $$\nabla f(x)+\frac{-...
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65 views

If a monotone function $f:[a,b]\to \mathbb R$ is differentiable on a closed interval $[a,b]$, is $f'$ bounded?

I first tried to find a counterexample $\sqrt{x}$ on $[0,1]$, but $\sqrt{x}$ is not differentiable at $0$. I also kwon the function $$f(x) \;=\; \begin{cases}x^2 \sin (1/x^2) & \text{if }x\ne 0, \\...
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54 views

Partial derivative of $f(x, y) = x ^ {x ^ {x ^ {x ^ y}}} + \ln(x)[\tan^{-1} (\tan^{-1}(\tan^ {-1}(\sin(\cos xy)-\ln(x+y)))]$

If $$f(x, y) = x ^ {x ^ {x ^ {x ^ y}}} + \ln(x)[\tan^{-1} (\tan^{-1}(\tan^ {-1}(\sin(\cos xy)-\ln(x+y)))]$$ Then what are the values of partial derivative $f_x(1,2)$ and $f_x(1,5)?$ I tried my best ...
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1answer
45 views

Differential Equation derivative question

I was given this example in class: $$\frac{dy}{dt}+\frac{y}{2}=\frac{1}{2}e^\frac{t}{3}$$ Finding $$\mu=e^\frac{t}{2}$$ After multiplying the whole equation we get: $$\frac{dy}{dt}\cdot{e^\frac{t}{2}}+...
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Series expansion for total derivative of finite order?

Given a function $L(t, x_1(t), \cdots, x_n(t))$, its total derivative with respect to $t$ is given by $$\frac{dL}{dt} = \frac{\partial L}{\partial t} + \sum_{j=1}^{n} \frac{\partial L}{\partial x_j} \...
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Derivative of piecewise function where partition subset depends on the variable with respect to which you are differentiating

How can we calculate $\partial u / \partial w$ with $$ u = \theta(\mu_R'w-\mu^*) $$ where $\theta$ is defined by the following piecewise function: $$ \theta = \begin{cases} \theta_1 & \mu_R'...
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Infinitly differenciable funcion on an open interval [closed]

Let $(a,b)\subset\mathbb{R}$ be an open interval. Let us define the function $f:\mathbb{R}\rightarrow\mathbb{R}$ so that $f\in C^\infty$ $x\in(a,b)\Rightarrow f(x)>0$ $x\in\mathbb{R}\setminus{(a,b)...
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What is the derivative of the inclusion map $\iota: M \rightarrow G \times_H M$

Let $G$ be a compact Lie group and $H$ be a Lie subgroup of $G$. Suppose that $M$ is a smooth manifold on which $H$ acts from the left. Let's consider the action of $H$ on $G \times M$ : $$h((g,m)):=...
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1answer
53 views

When is $\|f(x) - y\|$ a smooth function?

Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be a function and $\|\cdot\|_{\mathbb{R}^m}$ be the Euclidean norm in $\mathbb{R}^m$. Given a point $y\in\mathbb{R}^m$ what conditions does $f$ have to satisfy for ...
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1answer
36 views

using squeeze theorem to prove differentiability

I have this inequality \begin{align} 2x\leq f(x)\leq x^2 + 1 \end{align} and have to compute $\lim_{x\to 1^-} \frac{f(x)-f(1)}{x-1}$ to show that f(x) is differentiable at x = 1. I understand that I ...
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50 views

Whether the partial derivatives are bounded in any neighbourhood of $(0,0)$ or not.

The function is given as $f(x,y)=\begin{cases}(x^{2}+y^{2})\sin \left(\frac{1}{x^{2}+y^{2}}\right)& (x,y)\ne (0,0)\\0 &(x,y)=(0,0)\end{cases}$. The options are (a) The partial derivatives $\...
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vector derivative clarification

It has been a while and I can't remember the steps of how to get from line "J = " to line "dJ/dw". Any help would be appreciated thanks. Problem is here
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1answer
60 views

Integral of $2\int_0^{\infty} e^{-2y}e^{-y}{{(y)^x}\over{x!}} dy$

Is it possible to take this integral? $2\int_0^{\infty} e^{-2y}e^{-y}{{(y)^x}\over{x!}} dy$ I know I can use the fact that: $\int_0^{\infty} y^k e^{−y} dy = k!$ But I'm basically stuck on how to do ...
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25 views

Graphing $f$ from $f'$ (anti derivative) if $f'$ is an ECG curve

I was interested in the anti derivative of an ECG curve. The idea was electromagnetic induction occurs when current changes, and that what ECG might be reading is the derivative, the rate of change, ...
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26 views

Sign of Integral Derivative

$X_1$ and $X_2$ are iid draws from $(-\infty, \infty)$ according to $F$. Further let $A = a(x_1, \theta)/\cos (\alpha)$ and $B = (1- \cos(\alpha) - \sin (\alpha)(1-x_1))/\cos(\alpha)$, where $a(x_1,\...
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37 views

Power series with radius of convergence two.

Let $f(z)$ be a power-series (with complex coefficients) centered at $0 \in \mathbb{C}$ and with a radius of convergence 2 . Suppose that $f(0)=0$. Choose the correct statement(s) from below:\ $f^{-1}...
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1answer
41 views

Replacing right derivative with left derivative by replacing $x$ instead of $-x$

I was studying Cartan differential calculus and in section 3, there is a theorem that says: Assume $f:[a,b] \to F$ and $g:[a,b] \to R$ are continuous and for every $x \in (a,b)$ they have right ...
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1answer
85 views

In a certain sense, can "Piecewise Linear" be Interpreted as "Non-Linear"?

I was looking at the following function (called "ReLU") : I am trying to understand why this function ("ReLU") is considered to be non-linear, when it appears to look "...
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1answer
34 views

How do I determine further solutions of the equation using Rolle's theorem?

I gave this equation $2^x=1+x^2$ with the $1$st zero is $x_1=0$ and the $2$nd zero is $x_2=1$. (easy reading) Now I want to calculate more zeros using Rolle's theorem, and I've rearranged the function ...
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1answer
45 views

Proving formula for $n$-th derivative using l'Hopital (given existence of $n$-th derivative)

Let $f:J\rightarrow\mathbb{R}$ be $n$-times differentiable on an open interval $J$ ($f^{(n)}$ is not necessarily continuous) and $x\in J$. Then $$\lim_{h\rightarrow 0}h^{-n}\sum_{i=0}^n\binom{n}{i}(...
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1answer
16 views

Confusion in a step of the derivation of the invariance of Laplace's equation

I'm reading the derivation in Strauss's PDE book. He sets up the following relations: $$ r = \sqrt{x^2 + y^2 +z^2} = \sqrt{s^2 + z^2} , s = \sqrt{x^2 + y^2}, x = s \cos{\phi}, y - s \sin{\phi}, z = r ...
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86 views

I want to take a derivative of a given function

I have the function $$f(Z) := (-2\cos \theta_i(r_{Tx})L_y+\sum_{n=1}^N\{ (|\frac{\cos \theta_i(r_{Tx})Z_n-\eta_0}{\cos \theta_r(r_{Rx})Z_n-\eta_0}|^2)\cos \theta_i(r_Rx)) + {\bf{R}}[\frac{\cos \...
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1answer
26 views

Convex Functions Gradient Inequality : $f(x) \geq f(y) + \nabla{f(y)} \cdot (x-y)$

How do I prove that for a multivariate convex function $f:C\rightarrow \mathbb{R}$. Where $C$ is a convex set $f(x) \geq f(y) + \nabla{f(y)} \cdot (x-y)$ $\forall x,y \in C$
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1answer
52 views

A High School Calculus inequality

The problem, encountered in a high school math textbook in the exercises on the MVT,goes as following: Let $f(x)=e^x-ex, x\geq 0$ and $f(\ln{2})<2$, prove that $y=(1-e)x+1$ is tangent to the graph ...
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31 views

derivative of the sum

My book says: given \begin{equation} M_j\equiv e^{x_{jt}\beta-\alpha p_{jt}}+\xi_{jt} \end{equation} \begin{equation} s_{jt}=\frac{M_j}{1+\sum_{k=1}^{50}M_k} \end{equation} then \begin{equation} \frac{...
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1answer
42 views

How do I analytically check if a point is a local extremum when $f''(x)=0$?

If $f'(x_0)=0$ and $f''(x_0)=0$, how do I check whether or not it is a local extremum? The usual surefire method is to check the sign of $f'$ before and after $x_0$ , but I'm trying to prove a general ...
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1answer
63 views

Is a differentiable multivariable function with continuous derivatives on analytic paths continuously differentiable?

Let $f: \mathbb R^n \rightarrow \mathbb R$ be an everywhere differentiable function; and continuously differentiable when restricted to any analytic path. Is then $f$ continuously differentiable? I ...
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55 views

Area under graph and integration of dx

Integrating $dx$ within the bounds of $x_1$ and $x_2$ would result in $x_2-x_1.$ How does this make sense in terms of area under the graph?
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1answer
42 views

How to find the upper bound of an integral given the area

I am trying to do some decent programming for once, and it looks like math is unavoidable. I got pretty far with no math background, but now I'm stuck after days of trying to find an answer. I have ...
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0answers
35 views

Left side derivative [duplicate]

Let $(a,b)$ be an open interval in $\mathbb{R}$. $f:(a,b)\rightarrow\mathbb{R}$ $z\in (a,b)$ and let $f$ be continuous in $z$. Let $f$ be differentiable over $(a,b)\setminus\{z\}$. My question is ...
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0answers
31 views

Proof that derivative of a vector valued function is the derivative of its components [closed]

I just read about vector valued functions and their derivatives. It says that to compute the derivative of a vector function, just compute the derivative of its components. But i can't make myself ...
4
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3answers
157 views

What does the derivative really mean?

I was introduced to calculus a few weeks ago, and while I can "solve" problems consisting of derivatives and integrals, I still do not truly understand what the derivative means. Here are ...
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51 views

Are Saddle Points only Possible in Non-Convex Functions?

Informally, I have heard that the following are only possible in Non-Convex Functions: Saddle Points Local Minimums My Question: Is this in fact true - Can we mathematically prove that Saddle ...
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0answers
17 views

If a function to be maximized is almost everywhere differentiable can we say the derivative=0 condition must hold almost everywhere?

We have a known, increasing function $p:[0,1]\rightarrow[0,1]$. We want to find conditions on functions $R_1(\cdot),R_0(\cdot)$, $R_1(\cdot)$ known to be strictly increasing and $R_0(\cdot)$ known to ...
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1answer
41 views

Derivative of Softmax loss function with $\text{one}_\text{hot}(y)$ [closed]

Given the following: $h=g\left(w_1\cdot x+b_1\right)$ $ z=w_2\cdot h+b_2$ $y_\text{hat} = \mbox{softmax}\left(z\right)$ $\mbox{Loss}\left(y,y_\text{hat}\right)=-\sum \mbox{onehot}\left(y\right)\cdot \...
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0answers
47 views

Constant function and derivative [closed]

I have a doubt about the possibility to prove that $f$ is constant. I know that: let $f:(a,b)\to \mathbb{R}$ with $f'(x)=0$ $\forall x\in(a,b)$ then $f$ is constant. BUT if $(a,b)=\mathbb{R}$ this ...
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1answer
22 views

Chain rule for function of several variables

Say we have a function $$f:X\times Y \to \mathbb{R}$$ where $Y\subseteq \mathbb{R}^n$. I want to calculate $\frac{\partial}{\partial t}f(x,y_0+tu_i)$ for a fixed $y_0 \in Y$ and $u_i\in \mathbb{R}^n$ ...

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