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

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6 views

Exponents sum simplification problem

This might seem like a stats problem but it's actually a calculus issue. I'm trying to compute : $X'=argmax_X\{P(Y_0/X)P(X)\}$ Where $P(X)$ is a multivariate normal distribution and $P(Y_0/X)$ is a ...
5
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2answers
37 views

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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0answers
22 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
20 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
37 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
26 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
16 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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2answers
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 ...
1
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2answers
28 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
53 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)} ...
1
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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
19 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
14 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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0answers
17 views

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 ...
3
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1answer
24 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 ...
2
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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
40 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
24 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
44 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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0answers
23 views

Change of variable of integration, for numerical integration

I have an independent (array) variable $r = {r_0, r_1, ..., r_N}$, and three functions (arrays) of that variables, $n(r) ={n_0, n_1, ..., n_N}$, $p(r)$, and $E(r)$. How can I calculate the function ...
1
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1answer
33 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
201 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
20 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 ...
0
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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 } ...
1
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1answer
34 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 ...
3
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3answers
53 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 ...
0
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3answers
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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0answers
22 views

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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0answers
23 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
88 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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0answers
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 ...
2
votes
1answer
82 views

Derivative under integral mixed with…

$$f(x,y)=\int_{e^{4y}}^{\ln^3(x)}{\frac{\sin(t)}{t}\,dt}$$ Whats the derivative $\frac{d f}{d t}$, if: $$x(t)=\cos(2+6t).4t^2$$ $$y(t)=\ln(2r+7e^{5t})$$ Really not much to say about this problem ...
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4answers
75 views

Computing the sixth derivative of $F(x) = \int_1^x\sin^3(1-t)\mathrm dt$

Compute the sixth derivative at $x_0 = 1$ of $$F(x) = \int_1^x\sin^3(1-t)\mathrm dt$$ It's from a multiple choice test. I was able to narrow down the choices to $0$ and $60$. I guessed $0$ and ...
1
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1answer
36 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
2
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1answer
36 views

Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem Find derivative for $lnx^2$. It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha. I don't understand how to calculate the derivative for ...
1
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1answer
18 views

nth derivative of a troublesome function

I don't know where to start on this problem. I'm trying to get the 2015th derivative(at x = 0) of f(x) = x^2 * arctan(x). Doing the derivatives one by one seems a little troublesome... What do you ...