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

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Dependent and independent variables of a differential equation

$$\theta'' + \delta\theta'+\sin\theta = F\cos(\omega t)$$ I am trying to write it as a first order equation, and state the dependent, independent and parameters in the ODE. I have written it as: ...
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11 views

Mean value theorem sufficient conditions in several variables

I was doing a proof which was: If f is defined on an open set A, and all of its partial derivatives EXIST and are BOUNDED at A, then f is continuous. I used the trick of writng down (just to ...
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2answers
25 views

Explanation of the formula $df^{-1} = df\circ f^{-1}.$

Can someone explain the formula (for sufficiently nice $f$), $$df^{-1} = df\circ f^{-1}$$ So far, I have tried working with the relation $df^{-1} = (df)^{-1}$ and the chain rule but I am not able to ...
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1answer
12 views

Problems with differentiation of vector functions

This is really a homework problem for which I'd like a fresh set of eyes to look the solution over. The math isn't working out right. This means I've done something incorrect. Without further ado. ...
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1answer
35 views

how to proof that this function is zero

given $f$ continuous and diferentiable into $\mathbb{R}$ such that $\forall x\in\mathbb{R},|f'(x)|\le|f(x)|$ and $f(0)=0$ then proof that $f(x)=0$ atempt: taking $x>0$, since $f$ is ...
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2answers
35 views

Is there any geometric interpretation or significance of the complex roots of a derivative?

I was doing some reading online when I stumbled here and learned about this geometric way of viewing the complex roots of a function. It got me thinking; the zeros of the derivative of a function $f$ ...
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1answer
51 views

Tricky differentiation

$$\begin{aligned}f(x) = 1 - \left ( \frac{u-x}{u-l} \right )^{B}\end{aligned}$$ After differentiating, the answer is: $$\begin{aligned}f'(x) = \left ( \frac{B}{u-l} \right )\left ( \frac{u-x}{u-l} ...
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An example of a function for which the equality $M_1 = 2 \sqrt{M_0M_2}$ holds.

Let $f$ be twice differentiable on $(a,\infty),a\in \Bbb R$ and let $$M_k = \sup \{|f^k(x)|\mid x \in (a, \infty) \} < \infty, k=0,1,2.$$ $a)$ Prove that $M_1 \leq 2 \sqrt{M_0M_2}$. $b)$ Give an ...
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25 views

Find maximum area of triangle defined by tangent line to $y=e^{-x}$ [on hold]

Take a point $P(a,e^{-a})$ $(a>-1)$ on the curve $C:y=e^{-x}$. Let $S(a)$ be the area of the triangle surrounded by the tangent line to $C$ at $P$, the $x$-axis and the $y$-axis. (1) Find the ...
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1answer
22 views

Finding derivative of a split function using the definition of derivative

I have this function: $ f(x) = \begin{cases} \frac{sin^2(3x)}{x}, & \text{if $x\ne0$} \\ 0, & \text{if $x=0$} \end{cases} $ How would I find the derivative of it using the definition of the ...
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3answers
40 views

derivative of $\sqrt{kx}$

http://www.wolframalpha.com/input/?i=derivative+of+k^%281%2F2%29*x^%281%2F2%29 Here's my shot at it: $$f(x) = \sqrt{kx} = (kx)^\frac{1}{2}$$ $$f'(x) = \frac{1}{2}(kx)^\frac{-1}{2}$$ $$f'(x) = ...
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3answers
32 views

Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
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0answers
11 views

Is this the proper way to differentiate a vector with scalar?

$\vec{r}(t)=(r_0+kt)\cdot\begin{pmatrix}\sin(\omega t)\\\cos(\omega t)\end{pmatrix}$ $\vec{r}(t)=\begin{pmatrix}r_0\sin(\omega t)+kt\space \sin(\omega t)\\r_0\cos(\omega t)+kt\space \cos(\omega ...
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1answer
17 views

differential of inner product of functions from $R^n \to R^n$

I'm trying to find the differential of an inner product. Let $f:R^n \to R^n$ be $C^1(R^n) $ and let $x\in R^n,0 \neq v\in R^n$ . What is the derivative of $<f(x),v>$ ? If f was $R \to R^n $ ...
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29 views

Complex Analysis holomorphic function question

I have a Complex Analysis assessment question about holomorphic functions: Let f be a function on a plane and satisfies $f'(z) = f(z)$ and $f(0) = 1$ i) Give an example of a function with this ...
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1answer
53 views

Fundamental theorem of calculus, differentiable at the endpoints.

One version states: Let f be a continuous real-valued function defined on a closed interval $[a,b]$. Let f be the function defined for all x in $[a,b]$, by $F(x)=\int_{a}^xf(t)dt$. Then, F is ...
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2answers
30 views

How would I start to solve this?

I need to calculate the derivative of $F(x)=\int_{f(x)}^{f^2(x)}f^3(t)dt$. Usually for a derivative of an integral I would plug the upper bound and lower bounds into $f(t)$ then multiply each by their ...
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2answers
55 views

Prove that f is differentiable

Prove that if $f$ satisfies $|f(x,y)| \leq (x^2 + y^2)$ then $f$ is differentiable at $(0,0)$. I understand how to prove this: one can deduce that $f(0,0)=0$, and then we can assume that $L_{(0,0)} ...
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1answer
22 views

Assume that $f$ is a differential function and let $g(t) = \sin (f(-t,t,2t))$. Find $g'(t)$?

Assume that $f$ is a differential function and let $g(t) = \sin (f(-t,t,2t))$. Find $g'(t)$? I am confused, please help
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1answer
21 views

complex analysis differentiation and existence of a point?

If $f(z) = z^3$ prove that there is no point $c$ on line segment $[1,i]$ s.t. $(f(i)-f(1)) / (i-1) = f'(c)$. So differentiating: $$f'(c) = 3c^2$$ $$3c^2 = (f(i)-f(1))/(i-1) = (-i-1)/(i-1) = i$$ ...
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1answer
15 views

Rate of change question involving velocity, displacement and acceleration

I have been having trouble understanding questions c)-e) and am in need of some help: An object is moving in a straight line from a fixed point. The displacement $s$ in metres is given by ...
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1answer
45 views

What is the derivative of this? [duplicate]

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
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Partial integration and substition rule.

Well in my day to day usage I now came upon an example of using the substition rule where I can't see how it works, and I wonder how one could handle such an equation with ease. The set of equations ...
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1answer
25 views

Problem with the type of equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ and vertices?

I am asked to find the type equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ , represents ? i.e a parabola , or hyperbola or ellipse or circle by squaring twice? Now , what I have done is like this ...
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1answer
46 views

Can this special case happen when working with L'Hopitals rule?

I am using this version of L'Hopital's rule Assume that $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)=0$, and that the limit-value $\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$ exists (could ...
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1answer
42 views

Can we claim that $f(x)$ is an increasing function and its stationary point is $x = \infty$ (in this specific case)?

If $f(x)$ is concave and its first derivative $f'(x) \rightarrow 0$ when $x\rightarrow \infty$, can we claim that $f(x)$ is an increasing function and its (only) stationary point is $x = \infty$ ? ( ...
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1answer
26 views

is split function derivable

$ f(x) = \begin{cases} \frac{sin(x)}{x}, & x \ne0 \\ x+1, & x=0 \end{cases}$ I know that the function is a continuous function in R. But is this function derivable at x=0? I am not sure.. ...
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1answer
14 views

Estimating values using tangent line? [on hold]

How do you do this type of question and what would be correct answers in this case? Thank you all in advance!
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1answer
45 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
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1answer
35 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
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4answers
206 views

Chain rule for partial derivatives intuition

Can somebody give me an intuitive explanation for the above equations. I'm not sure how they come about and how they can be perceived logically. $$\frac{\partial z}{\partial s} =\frac{\partial ...
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3answers
21 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
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1answer
56 views

$ \exists c \in( a, b) \text{ such that } f(c)=\max\limits_{x \in [a, b]} f (x) $

I saw in a corrected. if We have $ f $ continuous on $ [a, b] $ with $ f (a) = f (b) $ and $ f $ differentiable left and right at $ (a,b)$ can we say that $$ \exists c \in (a,b) \text{ such that } ...
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1answer
35 views

nth derivative of ${1\over x}$. A problem. [on hold]

$f(x)=f^{(0)}(x)=x^{-1}$, $f^{(1)}(x)=-x^{-2}$, $f^{(2)}(x)=2x^{-3}$. Therefore, $f^{(n)}(x)=(-1)^{n}n!x^{-n-1}$. Except I see in some places that the expression is different, using, for example, ...
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1answer
16 views

Proving that $\sup f'\left( \left( 0,\infty \right) \right)=0$ under a certain set of conditions.

Let $f$ be a twice differentiable function on $\left( 0,\infty \right)$ s.t. $f''(x)>0$ for all $x\in \left( 0,\infty \right)$. Prove, that if the following conditions are satisfied: ...
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2answers
32 views

is it possible to intergrate this function to get x(t) and y(t)?

say you have a function as below; $d^2V(t)/dt = -B^2V(t)$ B is a constant Initial conditions $V_x(0) = V$, $V_y(0) = 0$ I can't see how to integrate to get x(t) and y(t); I ended up with ...
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3answers
54 views

Is the function complex differentiable at (0,0)?

(in Complex) $$ g(z) = \begin{cases} \frac{z^5}{|z|^4} & \text{if $z \neq 0$} \\ 0, & \text{if $z = 0$ } \end{cases} $$ For the function above, is it differentiable at $z=0$? I am trying to ...
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22 views

Finding maximum of convex function (appliance of derivatives)

The task goes as following: Divide the length of $14$ into parts $a$ and $b$, in a way that the sum of surfaces of two squares (which sizes are $a$ and $b$), is minimal. $14=a+b => b=14-a$ ...
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1answer
15 views

Derivative of a distribution

$\DeclareMathOperator{\vp}{v.p.}$ We define $\vp \frac 1x \in \mathcal D'(\mathbb R)$ (the principal value of $\frac 1x$) as $$\left\langle \vp \frac 1x, \varphi \right \rangle = \lim_{\varepsilon ...
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1answer
50 views

${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$

Let $f\in \mathbb{R}^{I}$ $I$ interval of $\mathbb{R}$ Show that $${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$$ in ...
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Relationship between supremum of the partial derivative and the derivative of the supremum

To fully set up the problem: I have a function $$F : \mathbb R^n \times \mathbb R^+ \to \mathbb R$$ such that $F$ is positive and for any fixed $t^* > 0$, $$F(x,t^*) \in H^\infty(\mathbb R^n);$$ ...
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3answers
40 views

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$.

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$. What I did so far: $f(x)=(x(2-x))^{-1}$. $f'(x)=-(x(2-x))^{-2}[2-2x]$ $f''(x)=2(x(2-x))^{-3}[2-2x]^2+2(x(2-x))^{-2}$. It confuses me a lot. I know I ...
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1answer
18 views

Finding the stationary points of a function

I have a question that I need help with. How do I find the stationary points of the following function? $$y = \frac{4x^3}{(x-1)^2}$$ I differentiated the function and got $$\begin{align} y' ...
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1answer
16 views

Questions on continuously differentiable function on $[a,b]$

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Normally we define derivatives of $f$ only at interior points in $[a,b]$. But when we write $f\in C^1([a,b])$, it means that $f$ is differentiable on ...
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24 views

Differentiation through the integral sign, more general case

I wondered in which cases, given a measurable space $(A,\mu)$, Banach spaces $E,F$, an open $U\subseteq E$ and $f:A\times U\rightarrow F$, we can conclude that the function $s\mapsto\int_A f(x,y)dx$ ...
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1answer
24 views

Differentiability implies continuous derivative? [duplicate]

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be ...
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2answers
26 views

Property of a differentiable function

Which one of the following is true: 1.If a function real valued function $f$ satisfies $|f(x)-f(y)|\leq |x-y|^{\sqrt2}$ for all $x,y\in \mathbb R$ is $f$ a constant? 2.If $f$ is ...
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1answer
89 views

Theorem $4.3.12$ on ( Mathématiques en BCPST Tome 1 Pascal BEAUGENDRE )

Let $f\in \mathbb{R}^{I}$ and $x_0 \in \stackrel{\ \circ}{I} = \mathrm{int}(A)$ Show that $$\left.\begin{matrix} f \text{ is continuous at } x_0 \\ f \text{ is differentiable at all } x \in I ...
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23 views

Derivative of $Ad(c(t))X$

Let $G=SO(3)$ and $V=\{c'(0)|c:(-\epsilon,\epsilon)\to G, c\in C^{\infty} , c(0)=1\}$. For $g\in G$, define $Ad(g): V\to V$ by $Ad(g)(X)=gXg^{-1}$. The book says ...
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2answers
22 views

Numerical Approximation for 2D Curvature

I have a list of points (x, y) that are taken from an unknown 2D parametric curve $\vec{f}(t)$. These points are monotonically increasing in t (ie: they're a "connect the dots" version of ...