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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2answers
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 ...
1
vote
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:
$\text {a) n}$
$\...
1
vote
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 ...
0
votes
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
$$\lim_{x \to e^a} \frac{\ln (1 + 7f(x)) - \sin (f(x))}{3f(x)}?$$
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
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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{...
3
votes
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 $$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{...
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1answer
25 views
If a function $y$ and its derivative $y'$ are defined on interval $I=[a,b]$,then does both $y$ and $y'$ are continuous on $I$?
I was reading a lecture slide and there it was written that since $y$ and $y'$ are defined on a closed interval I,they must be continuous on $I$. I tried to know the reason behind it. I thought that ...
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0answers
20 views
Notations for tensor products
So I'm in the need of writing some docs in latex on variable types of Tensor products, specifically involing third order derivatives tensor.
For first and second order derivatives there are standard ...
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vote
1answer
41 views
A chain rule for the angle
My question is fairly basic, namely can I do the substitution below?
$$
\frac{\mathrm{d}\theta(t)}{\mathrm{d}t} = -\frac{1}{\sin(\theta(t))}\frac{\mathrm{d}\cos(\theta(t))}{\mathrm{d}t}
$$
If not, ...
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votes
2answers
30 views
Creating a graph given condition of local minima and maxima
I had a question that goes like this:
Let $m$ be the number of local minima and $M$ be the
number of local maxima. Can you create a function where
$M > m + 2$ ? Graph.
I tried graphing it using ...
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votes
1answer
20 views
Inverse Matrix Differential
Suppose that we are given $g(\Sigma)=\Sigma^{-1}(\mu_0-\mu_1)$, where $\Sigma$ is p by p, and both $\mu_0$ and $\mu_1$ are p by 1.
Now I am hoping to find $\frac{dg}{d\Sigma}$. My current work is ...
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0answers
34 views
Trigonometric Proof using Differentiation
Find the largest value of $k$ such that $\tan x$ + $k$ $\sin x$ > (1+$k$) $x$ on the open interval (0, $\pi$/2). (Hint: $k$ is irrational)
Attempt:
Let $f$($x$) = $\tan x$ + $k$ $\sin x$ - (1+$k$) $x$...
-1
votes
0answers
20 views
derivative with respect to a function
I am a bit confused in evaluating the partial derivative:
I have this equation
$y(t)=x(t)+\dot{x}(t)\\$, where $\dot{x} = \frac{dx}{dt}$
$z=f(x,y,\dot{x})$ for example $z=\frac{(x-y)^2}{\dot{x}^2}$...
1
vote
1answer
28 views
Second “Total Derivative” of a Vector-Valued Function
I am working with a function $F : \mathbb{R}^3 \to \mathbb{R}^3$ and need to compute the vector of quadratic forms $Q$ given by $$Q(\textbf{u}) = \frac{1}{2} \textbf{u}^T\frac{d^2 F}{d\textbf{u}^2}(x,...
1
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0answers
33 views
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 ...
0
votes
3answers
32 views
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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votes
3answers
37 views
Minimization of Floor and Ceiling Functions
So the problem at hand is:
Find the minimum value of the following function for $ x> 0 $: $$ \def\lc{\left\lceil}
\def\rc{\right\rceil} \newcommand{\floor}[1]{\lfloor #1 \rfloor} x\lc x \rc + ...
5
votes
2answers
73 views
Prove $ \frac{d^n}{dx^n}\ln(x)=\frac{(n-1)!(-1)^{n-1}}{x^n} $ by induction
Prove
$$ \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} $$
which is ...
-1
votes
1answer
36 views
Find the derivative of $F(x) = \int_{\sin x}^{\cos x}e^{t^2+xt}dt$ at $x=0$
Let
$$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$$
But ...
0
votes
1answer
29 views
Maximum for function $(\theta-(\frac{\mu}{p-x})^a)·x$ [on hold]
Struggling with finding the maximum for this function:
$$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 ...
2
votes
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.
$...
1
vote
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(...
0
votes
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 ...
0
votes
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 ...
2
votes
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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votes
3answers
26 views
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).
0
votes
1answer
12 views
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 ...
1
vote
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 ...
0
votes
0answers
25 views
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,...
0
votes
2answers
27 views
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}$$
But I'm having trouble finding the points ...
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0answers
23 views
Derivative of Hadamard product with respect to matrix
I'm trying to calculate this derivative wrt matrix $F_{i}$ and simplify the whole expression:
$ \frac{ \partial J_{term 1}}{\partial \mathbf{F_{i}}}=
2 (\sum_{j:G(i,j)=1} (\mathbf{W}_{i,j} \...
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1answer
17 views
What are some practical applications of successive differentiation?
Before starting to learn something, I always wonder whats its application. So would you please give some practical examples of application of Successive differentiation and concepts related to it such ...
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0answers
16 views
Derivatives/Optimization, Macroeconomics-New Keynesian economy-from Heijdra book
So I get the process shown till (e).
What I do not understand is how they get from (f) to (g), and unfortunately, also from there till (j). If anyone could show me how they did the derivatives + ...
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0answers
12 views
Part of the proof of the first optimality condition. Showing that $f'(x*)=0$ for a local minimum $x*$.
Let $x*$ be a local minimum of a differentiable function $f(x)$, i.e. there exists $r>0$ such that for all $y \in B_n(x*,r)$ we have $f(y)\ge f(x*)$. Since $f$ is differentiable we have
$$f(y)=f(x*...
1
vote
2answers
35 views
Composite function derivatives
Is this statment right in derivative of compoaite three functions $(f \circ g \circ h)'(x) = (f\circ g)' (h(x))$
2
votes
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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vote
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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votes
1answer
28 views
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}{\...
1
vote
0answers
59 views
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 ...
0
votes
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 $j = \{ 0, 1, \ldots, N \}$ ...
1
vote
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 ...
1
vote
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 $f(c)=c.$
3
votes
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 ...
0
votes
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 ...
0
votes
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?
1
vote
2answers
29 views
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!!
