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

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2
votes
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 ...
0
votes
0answers
22 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 ...
0
votes
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
votes
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
votes
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: $$\dfrac {3x^{2}} {2(x-2)(x+2)}$$
0
votes
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
votes
2answers
523 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
votes
1answer
32 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
60 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
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 ...
0
votes
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 ...
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
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 ...
0
votes
0answers
11 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
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 ...
0
votes
3answers
85 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 ...
0
votes
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
vote
2answers
88 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
votes
1answer
58 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
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 ...
0
votes
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 = ...
2
votes
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 ...
11
votes
1answer
238 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
93 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
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
votes
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. ...
0
votes
0answers
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 ...
3
votes
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
0
votes
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 ...
1
vote
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 ...
0
votes
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= ...
0
votes
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
vote
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
votes
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. ...
0
votes
0answers
31 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
votes
3answers
26 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 ...
8
votes
1answer
57 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
178 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
22 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
31 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
24 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
53 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
60 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
117 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
38 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
21 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
61 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
38 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
43 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
104 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 ...