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

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Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2} $ where $C_1,C_2,f(x)$ has ...
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0answers
38 views

Derivatives, help please.

${E = - \cfrac{\sum\limits_{c}^C \log P(C)Y}{N}}$ where ${\log P(C) = I_c - \log \sum\limits_{c}^C \exp\left(I_c - \max_c\left(I_c\right) \right)}$ ${I_c = WO}$ ${W}$ is ${C \times N.Hid}$ weight ...
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2answers
27 views

Calculus Implicit Differentiation and Concavity

Consider the relation $4x^2 - y^2 = -2$ (a) Use implicit differentiation to calculate $dy/dx$ and find all critical points of the curve. (b) Calculate the second derivative and determine the ...
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2answers
46 views

Proof of application of Mean Value Theorem

Two bicyclists begin a race at 8:00AM. They both finish the race 2 hours and 15 minutes later. Prove/explain that at some point during the race, the bicyclists are traveling at the same velocity. So ...
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3answers
215 views

How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]

Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$ Is it possible to find $f'(x)$. If yes, please show all steps.
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1answer
61 views

Stuck trying to prove that $e^{-x^{-2}}$ is $C^{\infty}$ [duplicate]

This is Spivak's Calculus on Manifolds ex. 2-25, he says Define $f:\mathbb{R}\to \mathbb{R}$ by $f(x) = \left\lbrace \begin{array}{l} e^{-x^{-2}} &\text{ if } x \neq 0\\ 0 ...
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40 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
3
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4answers
89 views

Implicit Derivative approaches

Sorry for my excessive verboseness... Here's the equation as given: $$x = 10 + \sqrt{x^2 + y^2}$$ Here are my direct implicit steps without modifying original equation: $$\eqalign{ \dfrac{\mathrm ...
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0answers
35 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
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1answer
16 views

Find the equation of a line tangent at a specific point

I have to find an equation for the line tangent to the graph of $\large\frac {\sqrt{x}}{6x+5}$ at the point $(4,f(4))$, and write it out in the form of $y=mx+b$ Using the quotient rule I get.. ...
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3answers
54 views

surjective, but not injective linear transformation

$T$ is a transformation from the set of polynomials on $t$ to the set of polynomials on $t$. So, the input to $T$ should be a polynomial, and the output should be some other polynomial. Two common ...
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1answer
35 views

Differentiating Integrals

This problem appears as example 2d of Chapter 5 in "A First Course in Probability - Ross, 8th ed." Suppose that if you are s minutes early for an appointment, then you incur the cost cs, and if you ...
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3answers
63 views

Using the Chain Rule to prove trig derivatives

I'm having trouble with this problem, I'm not sure how to tackle it and I was wondering if somebody could set me on the right path. The problem is as follows: Use the Chain Rule to show that if ...
2
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1answer
71 views

Finding $\dfrac{d^nx}{dy^n}$

If $y$ is a function of $x$, then what is the relation between $\dfrac{d^nx}{dy^n}$ and $\dfrac{d^ny}{dx^n}$? If we were to talk about $\dfrac{dy}{dx}$ and $\dfrac{dx}{dy}$, then they both are ...
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1answer
29 views

Aftermath of Cauchy's mean value theorem

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 ...
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0answers
32 views

Uniform convergence result in proof of second-derivative formula

This is a fairly basic analysis question. Consider a continuous function $f: \mathbb{R} \to \mathbb{R}$ which is twice differentiable at a point $x$. If necessary, also assume that $f \in ...
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3answers
110 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 ...
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1answer
14 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 ...
2
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2answers
52 views

Find the derivative of $y=\cos(x) - 2\sin(x),$ when the gradient is $1$

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 ...
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0answers
26 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 ...
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1answer
44 views

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

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
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2answers
60 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} ...
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0answers
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Derive Chebyshev's inequality applying in finance [on hold]

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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3answers
60 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 ...
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1answer
53 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?
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2answers
32 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 ...
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1answer
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 $g(x) = m(x)\sin x$ ...
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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 ...
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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
84 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
307 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 $x=0$. ...
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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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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 ...
0
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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
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 ...
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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
30 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 ...
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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 ...
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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}) + ...
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
65 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 ...
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1answer
47 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 ...
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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 ...
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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
58 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} ...