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

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0
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3answers
81 views

What do we lose by differentiating without using the rules of differential calculus?

I learned differential calculus and its rules (quocient, chain, etc) and I got curious about one thing: What do we lose by not using these rules when differentiating? Obviously I've noted some utility ...
0
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0answers
8 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
3
votes
2answers
29 views

Find the derivative of y=cos(x) - 2sin(x), for gradient =1

I need to find the smallest positive value of $x$ for which the gradient of the curve has value 1. For this equation: $$ y =\cos(x)-2\sin(x) $$ The answer is 2.5c grad. The following is my ...
0
votes
0answers
12 views

Is this connected relation?

My task is to check if this is preference relation (connected and transitivited) $$ f \succeq g \Leftrightarrow \forall x\in [0,1] f'(x) \leq g'(x) $$ My solution is: that this relation is not ...
1
vote
1answer
41 views

Using 4 step-rule $y = 2/ (4t - 3)^{2}$ [on hold]

I tried solving it. My answer is $-4/16t^{2} + 48t + 18$, if your answer is different kindly show how is it done too thanks
3
votes
2answers
42 views

If $f'(x)\cdot x$ goes to zero then $f(2x)-f(x)$ is bounded.

Let $g:\mathbb R^m\to\mathbb R^n$ be defined by $g(x)=f(2x)-f(x)$ where $f:\mathbb{R}^m\to\mathbb{R}^n$ is a given differentiable function. The problem is to prove that if $\lim_{|x|\to\infty} ...
0
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0answers
12 views

Derive Chebyshev's inequality applying in finance

Hello because i am not familiar with this website, so i type my question as followed thank you for your helping.
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1answer
37 views

Trouble finding the derivative of an expression

I could use your help. I've spent over 20 minutes on this problem and my inability to solve it has my questioning my calculus skills. If someone could show me where I messed up and walk me through the ...
2
votes
1answer
3k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
0
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0answers
37 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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1answer
52 views

Anti derivative notation [duplicate]

$F$ is an anti derivative of $f$. $$\int f(x) dx = F(x)+C$$ Can you tell me why there is '$dx$' in the LHS?
2
votes
4answers
78 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 ...
-3
votes
0answers
41 views

Help solving a Problem [on hold]

The population of a particular city (in thousands) can be modeled by the function $$P(t)= \frac{500}{1+20e^{-0.05x}}$$ where x is the number of years after 1920. In what year was the growth rate of ...
1
vote
2answers
30 views

Proving double derivatives with the chain rule (I think?)

Hey StackExchange I'm having trouble understating where to start with this problem, I'm supposed to prove something about double derivatives and the chain rule but I'm having trouble understanding ...
2
votes
0answers
51 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 ...
0
votes
1answer
18 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 ...
0
votes
1answer
15 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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0answers
19 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 ...
1
vote
1answer
20 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 ...
5
votes
0answers
79 views
+50

Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of C1 functions on $[0,1]$ $f_n(x) \rightarrow 0$ for pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
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1answer
41 views

Calculus - Half Life [on hold]

A radioactive isotope has a half life of 30 years. If we started off with 10 mg of this isotope 12 years ago: What is formula to making an equation for this question?
4
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2answers
294 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 $x=0$. ...
0
votes
1answer
42 views

Calculus - Derivatives [on hold]

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
25 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 ...
3
votes
1answer
49 views

Vector by Matrix derivitive

According to wikipedia, there is no widely accepted definition of a Vector by Matrix derivative. I have a need of such a notion. For matrix w, and vector h. $$\mathbf{y=w \;h} $$ $$ ...
0
votes
1answer
27 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 $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...
1
vote
2answers
28 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 ...
0
votes
1answer
31 views
+50

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 ...
2
votes
2answers
40 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 ...
0
votes
1answer
45 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: $$\dfrac {3x^{2}} {2(x-2)(x+2)}$$
1
vote
1answer
33 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 ...
4
votes
1answer
282 views

Differentiation under the Integral Sign

Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space. Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions: 1.$f(x,y)$ is a measurable ...
2
votes
2answers
56 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 ...
0
votes
2answers
36 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
0
votes
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}) + ...
0
votes
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 ...
2
votes
1answer
75 views

Does the derivative of a bounded smooth monotone function have a limit at infinity?

Let $f \in C^1(\mathbb{R})$ a monotonic function such that $$\lim_{x \to \infty} f(x) = m \in \mathbb{R}$$ Does this imply $\displaystyle\lim_{x \to \infty} f'(x) = 0$? If so, can the hypothesis be ...
2
votes
1answer
544 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
0
votes
2answers
61 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
71 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
15 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 ...
2
votes
1answer
29 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
56 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
1answer
101 views

Calculus Optimization - Finding the minimum cost

In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land situations. A pipeline comes to a river that is 1 km wide at point A and must be ...
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
44 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
0
votes
2answers
35 views

Proving $u_x=u_y\Rightarrow f(z)=az+b$

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
38 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 ...
0
votes
0answers
22 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 = ...
0
votes
1answer
30 views

combination derivative and integration

could someone show me the step how to get this final answer \begin{align*} f(x) &= \frac d{dx} \, F(x)\\ &= \frac 1{B(a,b)} \frac{d}{dx} \int_0^{1-e^{-(\lambda x)^c}} x^{a-1}\, ...