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

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1answer
54 views

Question about Differentials

I am reading the book "Advanced Calculus" written by Kaplan, and here is what I have: Suppose that $y(x)$ is a differentiable function at $x = x_0$. Then, we can write $y(x_0+\Delta x) = y(x_0) + ...
0
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1answer
72 views

Construct a continuous function which has no derivative almost everywhere.

Georg Cantor is famous for the first set theory (in "naive" terms) and the diagonal argument. However Cantor is also credited with the Cantor Set and for constructing a continuous function which has ...
0
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1answer
61 views

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please ...
4
votes
1answer
86 views

Derivable doesn't exist in english?

I have a question about terminology. See this is what happens: someone says "this function is derivable", and then another, more experienced Anglo-Saxon mathematician goes on to correct this someone, ...
2
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1answer
44 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
2
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2answers
58 views

Trying to solve a Taylor series problem

I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$ Okay here goes: $$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$ ...
-3
votes
1answer
73 views

I am trying to find derivative $f$

I want to find the derivative of $f: [1, \infty] \to \mathbb{R}$ defined by formula $$f(x) = \int_0^{x^4} e^{t^2} dt$$ Here is what I have done: $F(b) = \int_0^b e^{t^2} dt$ and knowing ...
0
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1answer
45 views

Why we are checking differentiability in an open interval, why not in closed interval

When we check the differentiability of a function we will always take the domain to be an open interval.But the same definition holds for closed interval as well. Why we are considering ...
0
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2answers
42 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
2
votes
1answer
170 views

Find fourth derivative of function

The function is $\displaystyle{\frac{3x^4}{1-x}}$ and I am trying to find $\displaystyle{\frac{d^4}{dx^4}}$. However, I want to find the solution without using the quotient rule $4$ times in a row. I ...
0
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0answers
24 views

How to smooth the derivative of this function?

I've got a function of x $$\dot\gamma = \dot\gamma_0 exp\left(-\frac{\Delta F}{k \theta}\left[ 1 - |x|^p \right]^q\right)sgn(x)$$ which, for certain values of the constants and a range of x from about ...
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3answers
56 views

Pattern to ease differentiation.

Is there a way to differentiate the following without multiplying everything out? $$f(x)=\bigg(1 + \Big( 2 + \big(3 + (4 +x^6)^2~\big)^3~\Big)^4\bigg)^5$$ (Chain rule doesn't help much, binomial ...
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0answers
21 views

Operator differentiability

I was wondering, what techniques can one use to prove that an operator (let's say acting on real analytic functions and taking values in a Banach space) is infinitely differentiable? I know that, for ...
1
vote
2answers
70 views

Integrable function on $[0,2]$ and its antiderivative

I got this question: Let $f$ be the integrable function defined on the interval $[0,2]$ by the rule: $f(x)= \begin{cases} 4x^3 & \text{if $0 \leq x \leq 1$} \\ x^2+2 & \text{if $1<x \leq ...
1
vote
0answers
47 views

derivative of a matrix inverse

I wonder how to differentiate with respect to the diagonal matrix $X_d$, the following matrix : $$ X_d^T (\Sigma_d + X_d C X_d)^{-1} X_d $$ where $X_d$ and $\Sigma_d$ are diagonal matrices with ...
1
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1answer
43 views

Revisiting the product rule for derivatives

Let $E=C^{\infty}(\mathbb R, \mathbb R)$ Consider a linear transformation on $E$: $\delta$ such that $\forall f, g \in E, \delta(fg) =g\delta(f) +f\delta(g)$ Prove that there is some ...
1
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1answer
43 views

How do I prove continuity of partial derivative of $f(x, y) = \sqrt{x^4+y^4}$ at $(0,0)$?

Consider I have $$f(x, y) = \sqrt{x^4+y^4}$$ And I want to check if the function has partial derivatives continuous in point $$(x_0, y_0) = (0, 0)$$ I know theorem, that existence of continuous ...
2
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0answers
44 views

On continuous functions and second derivative

Let $f:[a,b]\to\mathbb R$ be a continuous function suh that $f''(x)$ exists $\forall x\in(a,b)$ . If $a<c<b$ and $f(a)=f(b)=0$ , then how to show that $\exists d\in(a,b)$ such that $f(c)=\dfrac ...
0
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1answer
43 views

Differentiation Matrix for central-difference scheme?

Central-difference scheme is defined to be: $f'(x) = \frac{f(x+d(x)) - f(x-d(x)))} {2*d(x)} + O(d(x)^2)$ Assume periodic boundary conditions, so that: $f(n+1)=f(1)$ I understand how to find all the ...
1
vote
2answers
70 views

Derivative of $\frac{\sin \coth x}{\csc \sqrt{e^{\log x}}}$

Derivative Problem: Hello, Ciao tutti, Buenos dias! I am trying find derivative with respect to x of function: $$ G(x)=\frac{\sin \coth x}{\csc \sqrt{e^{\log x}}}. $$ Derivativative rule for general ...
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2answers
40 views

Polar coordinates in the cartesian plane.

${dy}/{dx} = {dy}/{d\theta}$ divided by $dx/d\theta$ where $x$ and $y$ are in the Cartesian plane and $\theta$ is in the polar plane and $x = r\cos( \theta), \ y = r \sin (\theta)$. If $dy/dx = 0$ ...
0
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0answers
13 views

When is $c_1 \cdot f(g(x+c_2)) = f'(x)g(x)$?

We are allowed to pick and $c_1, c_2$ that helps make this question easier. So when is $$c_1 \cdot f(g(x+c_2)) = f'(x)g(x) \tag{1}$$ Also, separately, I'm wondering: $$c_1 \cdot f(g(x+c_2)) = ...
0
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1answer
45 views

Differentiation to Integration

Suppose we have the following relation: $$\frac{dx}{dt}=v$$ Then how does this imply the following: $$\int_{t_0}^{t} v\,dt=\int_{x_0}^xdx$$ The differentials cannot be treated as numbers since they ...
2
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2answers
45 views

An object is travelling in a straight line. Its distance, s meters, from a fixed point at time t seconds is given by the expression

$$s=t^3−t^2−6t$$ a) Find ds/dt when t=3 and interpret this result. b) Find d^2s/dt^2 when t=3 and interpret this result. c) Find the time in seconds when the velocity is 2m/s (d) Using the ...
0
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1answer
31 views

How to calculate the fastest change in value over time for all points in a set of data?

I have a list of data with the data represented as a tuple (Value, Time). For all of the data in my set I would like to calculate where the fastest rate of change ...
0
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2answers
64 views

How to find the derivative of $g(x)$?

I want to find this derivative, but I don't know what to do with the term $(x-t)^2$: Let $f:[0,1]\to\mathbb{R}$ be continuous. Define $g:[0,1] \to \mathbb{R}$ as follows: \begin{equation} ...
0
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1answer
29 views

Simple question about convergence and Gateaux derivative

If I consider the sequence $\{x_n\}\in L^2(\Omega)$ such that: $$ x_n \rightarrow x $$ We know that $x\in L^2(\Omega)$ because we're in a Banach space. So I can say that ...
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2answers
66 views

Values of $x>0$ of a curve

I got the following task: A curve has the equation $$ y = x^{\frac{3}{2}} + \frac{48}{x} $$ for values of $x > 0$. Find the coordinates of the turning point of the curve. By ...
0
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1answer
77 views

Why is the exterior derivative called exterior derivative

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ ...
0
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2answers
73 views

Proving that the function f is constant, mean value theorem, derivatives

Having the following inequality, for a real-valued function $f$ which is twice differentiable: $f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$. and assuming that $f$ is bounded, ...
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1answer
33 views

Using the chain rule for proof

how do you use the chain rule to show that: Thank you.
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0answers
114 views

how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$
0
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1answer
52 views

True rigorous meaning of dx symbol in general? [duplicate]

From Multivariate integration of a derivative w.r.t. a single variable, from words_that_end_in_GRY's answer. From elementary calculus, I always thought $\int^{}$ and $dx$ were just a kind of ...
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2answers
85 views

Directional derivatives in any direction are all equal

if a function has equal directional derivatives in all directions at a specific point, so is the function differentiable in it? I think it is correct, and i think if the function is f(x,y) so the ...
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1answer
650 views

deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
0
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1answer
26 views

computing directional derivate/ differentiability iff linear map exists

Let $\| \cdot \|$ be a norm on $\mathbb R^2$ and $S= \{ x \in \mathbb R^2 | \| x \| =1 \}$, $f: S \rightarrow \mathbb R$ a function with $f(-x) = -f(x)$ for all $x \in S$. Let $F: \mathbb R^2 ...
3
votes
4answers
202 views

What functions have the property that $\frac{d}{dx}f(x) = c \cdot f(x+1)$?

If we are allowed to pick any real-valued constant $c$ that helps, when does $$\frac{d}{dx}f(x) = c \cdot f(x+1)$$ In other words, when does the derivative of a function $f(x)$ equal some constant ...
2
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3answers
64 views

Limits of trig functions

How can I find the following problems using elementary trigonometry? $$\lim_{x\to 0}\frac{1−\cos x}{x^2}.$$ $$\lim_{x\to0}\frac{\tan x−\sin x}{x^3}. $$ Have attempted trig identities, didn't help. ...
0
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1answer
81 views

Help to solve this problem, the result doesn't seem right :/ silly mistake somewhere probably

Suppose that Coke and Pepsi are the only firms producing cola. Their products are not identical, but are very close substitutes. Let $P_c$ denote the price of Coke and $P_p$ the price of Pepsi. Demand ...
0
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1answer
420 views

Find the slope of the tangent line to the curve.

So I am trying to find the slope of the tangent line to the curve $$\sqrt{4x+2y} + \sqrt{xy} = \sqrt{38} + \sqrt{24}.$$ at the point $(8,3)$. I ended up implicitly differentiating and getting ...
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vote
1answer
104 views

Proof around Rolle's Theorem

Let $f(x)=\exp(\sin(2\pi x))$. I'm trying to prove that there is $a\in[0,1]$ such that, for $(n_r)$ a sequence of integers tending to $\infty$ (by this I mean $n_r$ tends to $\infty$ as $r$ tends to ...
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4answers
65 views

Differentiate $y=4\,e^{\cos2x}$

I do not know what to do for this question $$ y=4\,e^{\cos2x}$$ Can anyone show me or give me an example? I think I should be using the chain rule but not $100\%$ sure how to break up the question.
0
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1answer
35 views

Mean value of second differential

I have shown the first part that $\frac{d^2y}{dx^2}$ is that, in the second part they tell find the mean value of $\frac{d^2y}{dx^2}$ , how do I do this ? I don't understand what they mean or how to ...
2
votes
1answer
47 views

Smoothness of $f(x)/(1+|f(x)|)$ where $f \in C^1(E)$ for $E$ an open subset of $\mathbb{R}^n$

(a) Show that if $E$ is an open subset of $\mathbb{R}$ and $f \in C^1(E)$ then the function $$F(x) = \frac{f(x)}{1+|f(x)|}$$ satisfies $F \in C^1(E)$. (b) Extend the results of part (a) to $f \in ...
2
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1answer
40 views

Zero points of derivatives

It's obvious that if $f(x)$ is a polynomial then it's derivatives $f^{(n)}$ are equal to zero for $n>\deg f$. I'm trying to prove the "inverse" statement: if for each $x\in\mathbb{R}$ there ...
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1answer
20 views

Oscillating Spring & Rates of change

How to solve? Are they asking for: instantaneous rate of change: $\frac{d}{dt}h(t)=2.5$ and solve for value of $t$ or when $\frac{d}{dt}h(t_1)$ where $t_1$ is when $h(t)=2.5$ but both methods ...
2
votes
3answers
660 views

Differentiate the following function

$$y = \sqrt {\sin x} = (\sin x)^{\frac 12}$$ \begin{aligned} {dy \over dx} & = \frac 12 (\sin x)^{-\frac {1}2}{d\over dx} \sin x \\ & = \frac 12 (\sin x)^{-\frac 12} \cos x \\ & = ...
0
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1answer
44 views

Differentiate the following functions

Let $$y(x)= 4 x^3 e^{2x},$$ then $$y'(x) = 4 \times 3 \, x^2 e^{2x} + 4 \, x^3 \times 2 e^{2x} = 12 \, x^2 e^{2x} + 8 \, x^3 e^{2x}$$ Does this look correct?
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1answer
28 views

Simple partial derivatives and Chain Rule.

Shouldn't the function $g(x,y(x))=x^³ + y(x)$ satisfy $\frac{dy}{dx}=-\frac{g_x}{g_y}$? I get $g_x=3x^2 + \frac{dy}{dx}$ and $g_y=-1$ which does not satisfy it... Where do you think the mistake is? ...
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0answers
23 views

How to find the minima for $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$?

Please guide in how to find the value of $x$ for which $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$ will be minimum. I know this involves differentiation but am not sure on how to ...