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

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17 views

Discovering the derivatives of functions combined with trig values.

Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose $m(\frac{\pi}{3}) = 4$ and $ m'(\frac{\pi}{3}) = -2$, and let $g(x) = m(x)\sin x$ ...
0
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1answer
19 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
1
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1answer
21 views

Second derivative test of a function of two variables

From the following relation: How can we conclude the following rules: (i) Minima if both $f_{xx}$ and $f_{yy}$ are positive and $(f_{xy})^2 < f_{xx} f_{yy}$, (ii) Maxima if both $f_{xx}$ and ...
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0answers
93 views

Sign of the derivatives of a simple function

Consider the function $f(x)=x^b(1-x)^{1-b}$ defined on $[0,1]$, with $0 < b <1$. How can we prove that the even derivatives $f^{(2k)}$ have a constant sign on $(0,1)$? One can show that this ...
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2answers
308 views

Is it differentiable?

Let us consider the function $$ f(x)= \begin{cases} x^2\sin {\dfrac{\pi}{x}}\quad & x \neq 0\\ 0 & x=0 \end{cases} $$ We want to check its differentiability at $x=0$. ...
-1
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1answer
45 views

Calculus - Derivatives [closed]

Use the limit definition of a derivative $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ to show that the derivative of the curve $f(x)=4^x$ is $f'(x)=4^x\ln4$. [3 marks]
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2answers
27 views

Find the absolute maximum and absolute minimum values of f on the given interval

Find the absolute maximum and absolute minimum values of f on the given interval. $f(t) = t\sqrt{9 - t^2}$ on the interval $[-1,3]$. So $f'(x)=\frac{t}{2\sqrt{9-t^2}}+t\sqrt{9-t^2}$ and that is as far ...
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1answer
28 views

Two definitions for a smooth curve equal.

I've encountered these two definitions: 1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...
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0answers
20 views

Prove (non)differentiability in piecewise functions

I'm looking for some help on proving that this function is not differentiable at a specific value. My first instinct is to approach the limit of the value from positive and negative, but that doesn't ...
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2answers
69 views

Gradients and functions on matrices

Given a twice differentiable $f: \Bbb R \to \Bbb R$, with continuous second order derivative. We define $$F(x) = \sum_{i=1}^{m}f(x_i)$$ and $$L(x) = \sum_{i=1}^{m}f( \langle a_i, x \rangle+ b_i),$$ ...
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3answers
32 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
3
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2answers
45 views

About matrix derivative

Suppose $A$ is a matrix with order n*n. we have the following equity but I don't know why. $f(x)=\frac{1}{2}x^TAx-b^Tx$. then $f'(x)=\frac{1}{2}A^Tx+\frac{1}{2}Ax-b$ Is there any rule like scalar ...
1
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1answer
37 views

$\displaystyle k^{th}$ derivative of a Gaussian function with zero mean

The gaussian function is: $$f(x,\mu,\sigma)=\dfrac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\dfrac{(x-\mu)^2}{\sigma^2}\right)$$ Putting $\mu=0$, we can get the $\displaystyle k^{th}$ derivative of this ...
2
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2answers
59 views

What is the best way to find the derivative of binomials to a power? ((x+x^{-1})^3)'

I came to a problem on my homework and I want to know the best way to solve it. We are doing derivatives in Calculus. I've got the following: $$H(x)=(x+x^{-1})^3$$ $$H'(x)=((x+x^{-1})^3)'$$ I am ...
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2answers
33 views

Find the Derivative of fraction

I can't find out what I'm doing wrong again... $$f(x)=\frac{x^2+4x+3}{\sqrt{x}}$$ $$f(x)=\frac{x^2}{\sqrt{x}}+\frac{4x}{\sqrt{x}}+\frac{3}{\sqrt{x}}$$ $$f(x)=x^2(x^{9-1/2}) + ...
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4answers
81 views

Differentiating $\frac{te^{\tan t}}{ln(3t+1)}$?

I've tried to differentiate the following function: $$f(t)=\frac{te^{\tan (t)}}{ln(3t+1)}$$ But I am confused at what I should do (and perhaps I forgot some identities too), I've learned the ...
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0answers
21 views

derivative or differentiation with respect to a sum

I have the function $F(z',z,x,y)$, where $z=z(x,y)$ and $z'$ is the differential of $z$ with respect to its argument, and $x, y$ are the two independent varaibles here. So, $z$ and $z'$ are dependent ...
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2answers
66 views

Differential problem, how to get y''?

I've the following equation: $b^2x^2 + a^2y^2 = a^2b^2$, the first implicit derivative is: $\dfrac{dy}{dx} = \dfrac{-b^2x}{a^2y}$ I do not undertand how to find the second derivative of this ...
3
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2answers
73 views

Find $f'(0)$ given $f(x + y)$

Let $f$ be a differentiable function satisfying $$f(x + y) = e^xf(y) + e^yf(x)$$ for all $x, y \in \mathbb{R}$. Find $f'(0)$. I tried to use the definition of $f'(0)$ to do this: $$f'(0) = \lim_{h ...
0
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1answer
48 views

Derivative of a function with quotient rule: $\frac {3x^{3}} {2(x^{2}-4)}.$

The function is: $$\dfrac {3x^{3}} {2(x^{2}-4)}.$$ I'm using quotient rule: $$\dfrac{g(x)\cdot f'(x) - g'(x)\cdot f(x)}{{2(x^{2}-4)}^{2}}$$ The result i have is: $$\dfrac {3x^{2}} {2(x-2)(x+2)}$$
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1answer
17 views

Deriving marginal effects in multinomial logit model

For the multinomial logit model, it holds that: $$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$. Now my book states that the marginal effect is as ...
7
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2answers
521 views

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I ...
2
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1answer
31 views

What is a closed form expression for the ∂/∂w(∂t/∂w) if w(t) is complicated function?

Lets say we have a trigonometric function w(t) that can not be inverted as t(w). The derivative ∂t/∂w can be calculated as 1/(∂w(t)/∂t). What is a closed form expression for the second derivative ...
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2answers
59 views

Find the Derivative of $f(x) = 5t - 9t^2$

I'm stuck on this one: $$f(x) = 5t - 9t^2$$ $$f'(x) = \lim_{h\to 0} \frac{5(h+a) - 9(h+a)^2-5a-9a^2}h$$ $$f'(x) = \lim_{h\to 0} \frac{5(h+a) - 9(h^2+2ha+a^2)-5a-9a^2}h$$ $$f'(x) = \lim_{h\to 0} ...
0
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2answers
77 views

Why is this derivative 1/3?

My book asks me to find the derivative of: $$f(x) = \frac{1}{2}x - \frac{1}3$$ I'm trying to learn "the long way" apparently because we haven't learned the easy way yet, says my professor. The book ...
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2answers
42 views

If the real part of analytic function satisfies $u_x=u_y$, then the function is linear

Let f(z) be analytic function and $\forall z=x+iy\in\mathbb C, u_x=u_y$ ($u_x=\frac{\partial f}{\partial x},u_y=\frac{\partial f}{\partial y}$. Prove that $f(z)=az+b$. I thought using Cauchy ...
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2answers
42 views

$\dfrac {f(x)-f(0)}{g(x)-g(0)}=\dfrac {f'\big( \theta(x)\big)}{g'\big( \theta(x)\big)}$ , $\lim_{x \to 0+} \dfrac{\theta(x)}x=?$

$f,g:[0,1 ]\to [0,1]$ be continuous functions and twice differentiable in $[0,1]$ such that $g'(x) \ne 0 ,\forall x \in (0,1) , f''(0)g'(0) \ne f'(0)g''(0) $ , let $ \theta(x)$ be one of the numbers ...
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2answers
61 views

Help me check my homework

I have $$ h_\theta(x) = \frac 1 {1+e^{-\theta x}} $$ I need to get $ \frac d {d\theta} h_\theta(x) $. Here are my work. $$\begin{eqnarray} \frac d {d\theta} h_\theta(x) &=& \frac d ...
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0answers
10 views

Partial derivative in frequency domain when only time domain function is known

I want to calculate $$ \frac{\partial F_p(X(\omega))}{\partial X(\omega)} $$ So $F_p$ operates in some way on $X(\omega)$ but I know the analytical form only in time domain, represented by $f_p$. ...
1
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1answer
25 views

Variation on the Cauchy mean value theorem

From Spivak's Calculus, 4th edition, problem 11-50: Prove that if $f$ and $g$ are continuous on $[a,b]$ and differentiable on $(a,b)$, and $g'(x)\neq 0$ for $x$ in $(a,b)$, then there is some ...
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3answers
83 views

A strange (!) behaviour of differentiability

I see by drawing diagram that $y=\max (0,\sin x)$ is not differentiable at some points. But $y=(\max (0,\sin x)) ^ 2$ is . How to explain/prove it ? Am I missing something easy ? If $f$ is not ...
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1answer
35 views

Proving that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to z_0}\frac{f^\prime(z)}{g^\prime(z)}$

Let $f,g$ both analythic in neighbourhood of $z_0$ and they both have zero of multiplicity $n$ in $z_0$. Prove that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to ...
1
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2answers
85 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
5
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1answer
53 views

How to prove this inequality for $n$-th derivative: $\left|\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)\right|\leq \frac1n$?

It's easy to see that $\frac{d^n(\sin x)}{dx^n}=\sin \left(x+\frac{\pi n}{2}\right)$, so the following inequality holds: $$\left|\frac{d^n(\sin x)}{dx^n}\right|=\left|\sin \left(x+\frac{\pi ...
1
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1answer
33 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
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0answers
24 views

optimization problem with integrals

There is a maximization problem of the following form \begin{equation} \max_{l(a)} \sum \int \bigg(U(c, 1-l(a)) \bigg) x(a,e) da \end{equation} where $$ c = a(1+ f(L)) + e G(L)l(a) - h $$ $$ L = ...
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2answers
54 views

What do you get when you differentiate a $e^{f(x)}$-like function

I need help with exponential functions. I know that the derivative of $e^x$ is $e^x$, but wolfram alpha shows a different answer to my function below. If you, for example, take the derivative of ...
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1answer
235 views

Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of $C^1$ functions on $[0,1]$ $f_n(x) \rightarrow 0$ pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
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2answers
92 views

How to determine if it is l'Hopital's or not?

I got a lot of limits questions that I am able to find there limits, but I do not know if they meet the qualification to be l'Hopital's or not. SO how to know that ? For example: Q1) $ \lim_{t ...
2
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2answers
413 views

Derivative of $\frac1{1-x}$

Why is this not correct: $$ \frac{1}{1-x}= (1-x)^{-1} $$ now use chain-rule which gives: $(1-x)^{-2}$ times derivative of $(1-x)$ which is $-1$ so $$ -1\cdot (1-x)^{-2}= \frac{-1}{(1-x)^2} $$ why is ...
2
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1answer
36 views

Using the definition of a derivative

I'm here with a maths problem. I'm trying to find the derivative of an expression but when I calculate it I get different answers depending on if I use the power rule or definition of a derivative. ...
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50 views

Elementary integration and derivatives

Update Consider that the mean, of let's say a variable N is defined as: \begin{equation} N = E(e\,l) = \int\int e\, l(a) H(a,e) \end{equation} Where $E$ denotes the expected value (the random ...
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3answers
58 views

Finding the derivative of $\sqrt{x+\sqrt{x^2+5}}$

How to derive $y=\sqrt{x+\sqrt{x^2+5}}$ at $x=2$.I used logarithmic differentiation and chain rule over and over again but I can't get the right answer
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0answers
21 views

Differentiation of cost function in adaptive CFO estimator

I'me trying to simulate the steepest descent algorithm for CFO estimation using null subcarriers (OFDM wireless). And some mathematic difficulties have arised. In the core of algorithm lies cost ...
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3answers
70 views

Trouble finding the derivative of $\frac{4}{\sqrt{1-x}}$

I've been trying to figure out how to differentiate this expression, apparently I don't know my differentiation rules as much as I thought. I've been trying to use Wolfram Alpha as a guide but I'm at ...
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2answers
42 views

Derivative of y = $\sqrt{16x^2+5x+15}$

You are building a new house on a cartesian plane whose units are measured in miles. Your house is to be located at the point $(2,0)$. Unfortunately, the existing gas line follows the curve $y= ...
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0answers
36 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
1
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1answer
40 views

A Particular Frechet Derivative and Interpretation

I would like find the Fréchet derivative of the following functional: $$ \begin{align} F : C[0,1] &\rightarrow \mathbb{R}\\ w &\mapsto \frac{\int_0^1 xw(x)f(x) \, dx}{\int_0^1 w(x)f(x) \, dx}. ...
0
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2answers
24 views

Labeling derivatives of functions from a graph

I have a question about derivatives and identifying them on a graph. I came across a problem that looks like this: The figure shows graphs of f, f-prime, f-prime-prime, and f-pime-prime-prime. ...
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0answers
30 views

Reference for understanding Frechet and Gateaux derivatives

In multivariable calculus, when we were discussing directional derivatives, we were told that the fact that the directional derivative equals the gradient times the direction vector $( \partial^{\vec ...