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

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2
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1answer
34 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 ...
0
votes
2answers
88 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
votes
2answers
79 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 ...
0
votes
2answers
55 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 ...
0
votes
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 ...
1
vote
2answers
78 views

Differentiate $1/(1+e^{-\theta x})$ with respect to $x$

I have $$ h_\theta(x) = \frac 1 {1+e^{-\theta x}} $$ I need to get $ \frac d {d\theta} h_\theta(x) $. Here is my work. $$\begin{eqnarray} \frac d {d\theta} h_\theta(x) &=& \frac d {d\theta} ...
0
votes
0answers
25 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
vote
1answer
40 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 ...
0
votes
3answers
89 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 ...
1
vote
1answer
59 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
vote
2answers
133 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 ...
6
votes
1answer
65 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
vote
1answer
39 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 ...
2
votes
2answers
65 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 ...
11
votes
1answer
277 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 ...
2
votes
2answers
114 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
votes
2answers
419 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
votes
1answer
44 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. ...
3
votes
3answers
65 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
0
votes
0answers
26 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 ...
1
vote
3answers
104 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 ...
0
votes
2answers
48 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= ...
0
votes
0answers
38 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
vote
1answer
53 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
votes
2answers
41 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. ...
0
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0answers
36 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 ...
0
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3answers
27 views

Let $S(x)=\sum\limits_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}$. Define $C(x):=S'(x)$ and show that $C'(x)=-S(x)$.

so this problem I am trying to solve says Let $S(x)=\sum\limits_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}$. Define $C(x):=S'(x)$ and show that $C'(x)=-S(x)$. Well, I get that ...
7
votes
1answer
60 views

How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?

How can I find $f(0)$, $f'(0)$, and $f'(x)$ given that $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$.
2
votes
3answers
206 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
1
vote
1answer
25 views

Reference for a property of convex function

Let $F:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. It is known that if $F$ has a strict local maximum, then it is not a convex function. I just would like to ask you for a ...
0
votes
2answers
33 views

Maximum value of a function using derivatives

For which x does the function $f(x) = x^3-6x^2-5x+5$ assume its maximum value on the interval $[-5,5]$? The critical points for this function are $\frac{12 + \sqrt{204}}{6}$ and $\frac{12 - ...
0
votes
1answer
26 views

Radius of curvature and continuous functions

Let $\kappa (x)$ be radius of curvature function for a continuous function $f(x)$. Is it necessary that $\kappa(x)$ will have extrema when $f(x)$ does. And the nature of extrema is opposite to that ...
0
votes
3answers
55 views

Derivative of $f(0)=0$

Why is the derivative of $f(x) = x-(x^2-2x)$ not defined at $x= 0$? For a function $f(x) = x-|x^2-2x|$, the differentiation is possible when is broken into a piece wise function. i.e. $$f(x) ...
0
votes
1answer
66 views

Arc Length with Vector-Valued Functions

"Consider the path of a particle in a conservative force field represented by the vector-valued function $r(t) = \langle 4(\sin t - t \cos t), 4(\sin t + t \sin t), (\frac{3}{2})t^2 \rangle$." "A) ...
2
votes
4answers
161 views

Derivative conundrum…

I've spent almost 6 total hours hacking at this problem. And I always end up by a factor of 3 in one of the terms when checked against Wolfram's derivative calculator, which is correct when I manually ...
0
votes
1answer
40 views

Calculus Review - Differentiating an Integral

I'm trying to review some calculus over the summer and I just wanted to double-check my answer to a simple problem I came up with myself. Thanks. What is: $\frac{d}{dx} \int_a^{g(x)} f(t)\;dt\;$? ...
2
votes
1answer
39 views

Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what ...
0
votes
2answers
74 views

Is my theorem correct? $f(x) \leq g(x)$ for $x\geq a$ iff $f'(x) \leq g'(x)$ for $x\geq a$ and $f(a)=g(a)$.

I am trying to invent a theorem by inspection, which is $f(x) \leq g(x)$ for $x\geq a$ iff $f'(x) \leq g'(x)$ for $x\geq a$ and $f(a)=g(a)$. Is it correct?
0
votes
0answers
48 views

Cauchy–Riemann equations

What are the steps to find many functions that satisfy Cauchy–Riemann equations at a point $$z=z_0$$ but are not differentiable at that point
1
vote
2answers
45 views

On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...
1
vote
1answer
121 views

Derivatives in the real world

Two row boats start at the same location, and start traveling apart along straight lines which meet at an angle of $\pi/3$. Boat A is traveling at a rate of $10$ miles per hour directly east, and boat ...
0
votes
1answer
39 views

Comfirmation of third derivative of symbolic equation including summation

With previous help I was able to find the first derivative of an equation for a work project. Now I'm after the second and third derivative, for use in a program to find the maximum (Which I must do ...
1
vote
2answers
241 views

Related rates calculus problem involving shadow lengths

A light on the ground is 30 feet away from a building. A 4 foot tall man is walking from the light to the building at a rate of 3 feet per second. He is casting a shadow on the side of the building. ...
1
vote
1answer
53 views

Extrema and inflection points of the function $y = \cos^2(x) - \cos(x)$

Please help me find the extrema and inflection points of the function $y = \cos^2(x) - \cos(x)$. So far: $$y'=-2 \cos(x)\sin(x)+\sin(x)$$ $$y'' = 2\sin^2(x) - 2\cos^2(x) + \cos(x)$$ $y' = 0$ when $x ...
0
votes
2answers
61 views

If $foo=f(x,\dot{x})$, what are $\frac{\partial f}{\partial \dot{x}}$ and $\frac{\partial f}{\partial x}$?

Question Let $x$ be a function of $t$ and $f=f(x,\dot{x})$. What are $\frac{\partial f}{\partial \dot{x}}$ and $\frac{\partial f}{\partial x}$ ? Example For example if ...
0
votes
3answers
145 views

Why $2x$? Can't it be $x$? [duplicate]

So today in my school our neighbor class monitors were complaining to that few of our students were yelling and making noise. Actually the case was that we were having very aggressive debate over a ...
5
votes
2answers
139 views

Can the limit $\lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ exist if $f'(x)$ does not exist at $x$?

The second derivative of $f$ can be written as $$f''(x) = \lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$$ while it can also be written as (in fact, I believe this is the definition of ...
5
votes
5answers
116 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
0
votes
1answer
41 views

Derivative of two-variable exponential function [closed]

How do you get the derivative of this thing? $f(x,y)=20x^{3/2y}$
-1
votes
1answer
54 views

Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...