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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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23 views

higher derivatives

Use the product rule three times to find a formula for (fg)''' and compare the result with the expansion (a+b)3. Then try to guess a general formula for (fg)(n). I don't really understand what they ...
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1answer
62 views

Let $f(x)$ be a $n$ degree polynomial function having $n$ real and distinct roots. If $g(x) = f'(x) + 100f(x)$

Let $f(x)$ be a $n$ degree polynomial function having $n$ real and distinct roots. If $g(x) = f'(x) + 100f(x)$, then minimum number of roots that $g(x)$ must possess is: $\text {a) n}$ $\...
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1answer
45 views

What is $f'(0)$ for $f(x) = x^{1/3} \sin x$? Does the product rule give a different answer?

If $f(x)= x^{1/3} \sin x$. Say we want to find the derivative using the rules of differentiation. So $f'(x)=\sin(x) x^{-2/3} + \cos(x) x^{1/3}$, and we see that $f'(0)$ is undefined. But if we use ...
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1answer
36 views

Application of derivatives problem [on hold]

The graph of the function $y = f(x)$ has a unique tangent at $({e^a},0)$ through which the graph passes. What is $$\lim_{x \to e^a} \frac{\ln (1 + 7f(x)) - \sin (f(x))}{3f(x)}?$$
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1answer
30 views

How to find a function that is defined by its integral and its derivative?

I was trying to solve a physics problem. The question is: "How long will it take for a boat to cross a river if its velocity is always directed towards a fixed point on the opposite side of the river ...
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2answers
24 views

Find $ h'(4)$, in terms of $f(4)$ , $f '(4), g(4),$ and $g '(4).$

The following equation is a function: $$h(x)=\frac{f(x)+g(x)}{x}$$ Find $ h'(4)$, in terms of $f(4)$, $f '(4)$, $g(4)$, and $ g '(4)$ using ONLY Power rule, Product rule, or Quotient rule. I ...
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2answers
25 views

Factorising functions out of partial derivatives

I have been doing that work that requires me to use the chain rule on second order partial derivatives to replace variables (x, y) with (u, v) where u and v are functions of x and y. My question is ...
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2answers
52 views

Is this function always differentiable?

I need some help with the following exercise: Assume $A \subset \mathbb{R}$ is a compact set. Is the following function always differentiable? $f: \mathbb{R}^{n+1} \rightarrow \mathbb{R}, \quad f(...
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1answer
31 views

Derivation Numerical Method with partial derivatives, vectors, matrices and scalar product

I need help in finding a way to combine the equations \begin{equation} \frac{\partial J}{\partial W} \cdot \delta W = \langle Y_M^T (\eta^{'}(Y_M W) \odot (\eta(Y_MW)-C)),\delta W \rangle \end{...
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1answer
38 views

Contour integration and the central binomial coefficients

I am trying to compute the integral $$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{n + 1}}\ dx.$$ From computational evidence, it's very obvious that $$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{...
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1answer
25 views

If a function $y$ and its derivative $y'$ are defined on interval $I=[a,b]$,then does both $y$ and $y'$ are continuous on $I$?

I was reading a lecture slide and there it was written that since $y$ and $y'$ are defined on a closed interval I,they must be continuous on $I$. I tried to know the reason behind it. I thought that ...
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0answers
20 views

Notations for tensor products

So I'm in the need of writing some docs in latex on variable types of Tensor products, specifically involing third order derivatives tensor. For first and second order derivatives there are standard ...
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1answer
41 views

A chain rule for the angle

My question is fairly basic, namely can I do the substitution below? $$ \frac{\mathrm{d}\theta(t)}{\mathrm{d}t} = -\frac{1}{\sin(\theta(t))}\frac{\mathrm{d}\cos(\theta(t))}{\mathrm{d}t} $$ If not, ...
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2answers
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Creating a graph given condition of local minima and maxima

I had a question that goes like this: Let $m$ be the number of local minima and $M$ be the number of local maxima. Can you create a function where $M > m + 2$ ? Graph. I tried graphing it using ...
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1answer
20 views

Inverse Matrix Differential

Suppose that we are given $g(\Sigma)=\Sigma^{-1}(\mu_0-\mu_1)$, where $\Sigma$ is p by p, and both $\mu_0$ and $\mu_1$ are p by 1. Now I am hoping to find $\frac{dg}{d\Sigma}$. My current work is ...
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Trigonometric Proof using Differentiation

Find the largest value of $k$ such that $\tan x$ + $k$ $\sin x$ > (1+$k$) $x$ on the open interval (0, $\pi$/2). (Hint: $k$ is irrational) Attempt: Let $f$($x$) = $\tan x$ + $k$ $\sin x$ - (1+$k$) $x$...
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derivative with respect to a function

I am a bit confused in evaluating the partial derivative: I have this equation $y(t)=x(t)+\dot{x}(t)\\$, where $\dot{x} = \frac{dx}{dt}$ $z=f(x,y,\dot{x})$ for example $z=\frac{(x-y)^2}{\dot{x}^2}$...
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1answer
28 views

Second “Total Derivative” of a Vector-Valued Function

I am working with a function $F : \mathbb{R}^3 \to \mathbb{R}^3$ and need to compute the vector of quadratic forms $Q$ given by $$Q(\textbf{u}) = \frac{1}{2} \textbf{u}^T\frac{d^2 F}{d\textbf{u}^2}(x,...
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Is there a trick to compute this multinomial-looking sum?

The series I want to sum has this form $\displaystyle \sum_{} 1^{l_1}(1+c)^{l_2} (1+2c)^{l_3} \cdot \ldots \cdot (1+(N-1)c)^{l_{N}}$ for some constant $c$ and positive integers $N$ and $L$. Here ...
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3answers
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Cannot understand the straight line equation in Newton Raphson method.

I'm currently trying to gain an intuitive understanding of the Newton Raphson method but have reached a hurdle I seem unable to jump at the moment: Here's where I am so far: We have a function $f(x)$...
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3answers
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Minimization of Floor and Ceiling Functions

So the problem at hand is: Find the minimum value of the following function for $ x> 0 $: $$ \def\lc{\left\lceil} \def\rc{\right\rceil} \newcommand{\floor}[1]{\lfloor #1 \rfloor} x\lc x \rc + ...
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2answers
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Prove $ \frac{d^n}{dx^n}\ln(x)=\frac{(n-1)!(-1)^{n-1}}{x^n} $ by induction

Prove $$ \frac{d^n}{dx^n}\ln(x)=\frac{(n-1)!(-1)^{n-1}}{x^n} $$ by induction. Attempt to solve Base case $n=1$ $$ \frac{d}{dx}\ln(x)=\frac{(1-1)!(-1)^{1-1}}{x^{1}}=\frac{1}{x} $$ which is ...
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1answer
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Find the derivative of $F(x) = \int_{\sin x}^{\cos x}e^{t^2+xt}dt$ at $x=0$

Let $$F(x) = \int_{\sin x}^{\cos x}e^{t^2+xt}dt$$ How to find $F'(0)$? The answers says $$F'(x) = e^{\cos^2x+x\cos x}(-\sin x)-e^{\sin^2x+x\sin x}\cos x+\int_{\sin x}^{\cos x}te^{t^2+xt}dt$$ But ...
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1answer
29 views

Maximum for function $(\theta-(\frac{\mu}{p-x})^a)·x$ [on hold]

Struggling with finding the maximum for this function: $$f(x)=\Bigl(\theta-\Bigl(\frac{\mu}{p-x}\Bigr)^a\Bigr)·x$$ where $\theta>0, \mu>0, p>0, \alpha>1$. Wolfram Alpha gives me the ...
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1answer
59 views

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and $f$ is differentiable at z.

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and f is differentiable at $z$. The $\bullet$ denotes the dot(inner) product. $\nabla$ is the gradient. $...
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1answer
41 views

Derivative of a trace of quadratic form

I think the question is close to Derivative of trace of inverse of a matrix function I have a function: $f(X) = trace(X^TAX)$ And I am trying to derive $df/dX$ Update: I know that, for function $g(...
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1answer
27 views

Find derivative without the function, only results

If $g(4)=5, g’(4) = \frac23$, obtain derivative of $f^{-1} (x)$ at $x = 5$. I’m lost on how I can get this, I tried $\frac{g’(x)}{x} but$ confused to how to get a function to plug in $5$. Am I ...
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1answer
34 views

A derivative of the inverse of the function

Consider $f(x) = x^3 - 3x^2 - 1$ where $x \geq 2$. Find derivative for $f^{-1}(x)$ at point where $x = -1 = f(3)$. I tried getting $f^{-1}(x)$ and then getting the derivative of that but it th ...
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2answers
47 views

Prove $f(x)=x^n(1-x)\lt \frac{1}{ne}$ for all $n\in \mathbb{N},x\in(0,1)$

Prove $f(x)=x^n(1-x)\lt \frac{1}{ne}$ for all $n\in \mathbb{N},x\in(0,1)$. My try: $f'(x)=nx^{n-1}-(n+1)x^n=x^{n-1}(n-(n+1)x)=0\Rightarrow x=\frac{n}{n+1} $, hence $\max_{x\in(0,1)} f(x)=f(\frac{n}{n+...
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3answers
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Is $c \frac{\partial}{\partial x}=\frac{\partial c}{\partial x}=0$?

Is $c \frac{\partial}{\partial x}=\frac{\partial c}{\partial x}=0$? Or is one supposed to treat $c \frac{\partial}{\partial x}$ as "partially applied" operator (which expects a function).
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1answer
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Derivative and extremum

I try to solve a problem but I just can't figure it out: I have this equation as a statement: $f(x)= x^4+ax^3+bx^2+cx+d$ There is a maximum at $x=0$ There are four propositions and one is supposed ...
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2answers
56 views

Are all derivatives vectors?

Before reading this question, let me just state that prior to this year, I had allways thought vectors were arrows in space or lists of numbers, and I am still getting accustomed to their more formal ...
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discontinuities in first derivatives of PDEs

In the lecture notes (Oxford University, Applied Partial differential equations) it says that $u$ is continuous and only first derivatives of $u$ may be discontinuous across some curve $C$ in the $(x,...
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Problems with finding horizontal and vertical tangents for this equation $3(x^2+y^2)^2=100xy$

Our professor gave us this function to differentiate $$3(x^2+y^2)^2=100xy$$ and I did differentiate it $$\frac{dy}{dx}=\frac{3x^4+3xy^2-25y}{25x-3x^2y+3y^3}$$ But I'm having trouble finding the points ...
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0answers
23 views

Derivative of Hadamard product with respect to matrix

I'm trying to calculate this derivative wrt matrix $F_{i}$ and simplify the whole expression: $ \frac{ \partial J_{term 1}}{\partial \mathbf{F_{i}}}= 2 (\sum_{j:G(i,j)=1} (\mathbf{W}_{i,j} \...
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1answer
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What are some practical applications of successive differentiation?

Before starting to learn something, I always wonder whats its application. So would you please give some practical examples of application of Successive differentiation and concepts related to it such ...
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Derivatives/Optimization, Macroeconomics-New Keynesian economy-from Heijdra book

So I get the process shown till (e). What I do not understand is how they get from (f) to (g), and unfortunately, also from there till (j). If anyone could show me how they did the derivatives + ...
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Part of the proof of the first optimality condition. Showing that $f'(x*)=0$ for a local minimum $x*$.

Let $x*$ be a local minimum of a differentiable function $f(x)$, i.e. there exists $r>0$ such that for all $y \in B_n(x*,r)$ we have $f(y)\ge f(x*)$. Since $f$ is differentiable we have $$f(y)=f(x*...
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2answers
35 views

Composite function derivatives

Is this statment right in derivative of compoaite three functions $(f \circ g \circ h)'(x) = (f\circ g)' (h(x))$
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4answers
28 views

Linear function ortogonal at each point is a cross product

Let $f\colon \mathbb{R}^3 \to \mathbb{R}^3$ be a linear function satisfying $$\langle x, f(x) \rangle = 0$$ for all $x \in \mathbb{R}^3$. I want to prove (and hopefully not disprove) that there ...
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5answers
125 views

A Special Property of e?

I was debating whether I should post this question over at MathEducatorsStackExchange or here, sorry if this was the wrong forum for this question. I am teaching Calculus I this semester and I found ...
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1answer
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Step in the proof of the adjoint representation of the Lie bracket

Let $a: \mathbb{R}^2 \rightarrow M$ a differential map such that $a(s, 0) = p$ for all $s \in \mathbb{R}$. Let $\gamma : \mathbb{R} \rightarrow T_p M$ the path given by $\gamma(s) : = \frac{\partial}{\...
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0answers
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Differentiation under the integral $\frac{d}{dt}\int_0^s cos(f(x,t))\,dx$

I'm trying for a while to solve this problem, but unfortunately without any success. Assuming that the two functions $A$ and $f$ are known at time $t$ for the different positions $x$ (between $0$ and ...
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1answer
25 views

Discrete derivative formulation by Taylor Expansions

I'm following the paper "Cordova 2014, Comparative Study of two compact finite difference methods". It states: Given a discretization of a line by $x_j = -1 + jh$, where $j = \{ 0, 1, \ldots, N \}$ ...
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3answers
34 views

Find the cubic polynomial f(x) such that

Find the cubic polynomial f(x) such that $𝑓(−1)=-4$, $𝑓'(−1)=4$, $𝑓''(−1)=-6$, $𝑓'''(−1)=12$ I know that i have to use an augmented matrix in order to solve it, so i decided to use the ...
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1answer
31 views

Mean Value Theorem ( i guess )

Suppose that a function $f$ is continuous on the closed interval $[0,1]$ and that $0\leq f(x)\leq 1$ for every $x \in [0,1]$. Show that there must exist a number $c$ such that $f(c)=c.$
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1answer
51 views

Inequality on the exponential function

By playing around, I seem to have come across the following inequality, valid for all $x$: $$x-(1-e^{-x}) \ge e^{-\frac{2}{x}} x$$ (The constant $2$ is not necessarily the tightest one possible.) Is ...
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4answers
46 views

Maximize area of triangle

I'm trying to maximize the area of a triangle with the following three sides. The first side is the y=0, the second lies on the line y = 3x, and the third passes through the point (1,1). I want to ...
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0answers
36 views

Why is derivative different

$y^2 = (x-1)/ (x+1)$ implicates that derivative is $y'= 1/y(x+1)^2$. But if you eliminate denominator, then one obtains $y' = (1-y^2)/2y(x+1)$. Why is this different?
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How should I use the alternate formula of the derivative to “mathematically” show that the derivative of $f(x)=|x-2|$ at $x=2$ does not exist?

The title is the question. But I’m not sure how one can “mathematically” show the derivative? Can someone please interpret that part of the problem? Thanks!!