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

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70 views

How do you differentiate -|t|?

How do you differentiate $-|t|$? Using Wolframalpha it says to re-write $-|t|$ as ($-\sqrt{t^2}$). Why? (This is part of a bigger question, that being to calculate the differential of $e^{-|t|/T}$ ).
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1answer
25 views

What is the value of $[c,d]$ when $c$ and $d$ be such that $f(x) ∈ [c, d]$ for all $x ∈ [a, b]$?

Let $c$ and $d$ be such that $f(x) \in [c, d]$ for all $x \in [a, b]$. What is the value of $[c,d]$ for the function $f(x)=\sqrt{1-x^2}$ on the interval $[a, b]=[0,1]$? I knew taking the minimum and ...
1
vote
1answer
95 views

Differentiability Theorem Question

$f(x,y) = \begin{cases} \frac{1}{2} y \log(x^2+y^2), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ You may assume that this is a continuous function. Prove that f does not satisfy ...
3
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1answer
94 views

Math Differentiation Limits

$$\lim_{x\to 3} \frac{2x^2 + 7x-15}{x-3}$$ What I Simplified Step 1 : $\frac{2x^2 + 10x -3x -15}{x-3}$ Step 2 : $\frac{2x(x + 5)-3(x + 5)}{x-3}$ Step 3 : $\frac{(2x - 3)(x + 5)}{x-3}$ but unable ...
2
votes
2answers
472 views

derivative with respect to $\log(x)$

I have a dynamic equation, $$ \frac{\dot{k}}{k} = s k^{\alpha - 1} + \delta + n$$ Where $\dot{k}/k$ is the capital growth rate as a function of savings $s$, capital $k$, capital depreciation rate ...
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votes
2answers
121 views

Taking derivatives of exponential function

Beware, this question might be silly and may contain mathematical fallacies. $$ d/dt(e^{jwt}) = jwe^{jwt} $$ $$ d/dt(e^{j \pi t}) = j \pi e^{j \pi t} $$ $$ d/dt(e^{j 180 t}) = j 180 e^{j 180 t} $$ ...
7
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2answers
211 views

A (not so?) simple calculus problem

I'm the teaching assistant for a first semester calculus course, and the professor has given the students the following problem: Find the points on the curve $xy=\sin(x+y)$ that have a vertical ...
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3answers
61 views

Use the limit of the derivative to find the slope of the tangent line to the graph of $y=1/x^2$ at the point $(1,1)$

I'm having major difficulty understanding derivatives and limits. I need help.
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2answers
846 views

Find the position of the object when the velocity is 0

The position of an object given by $f(t)=5t^2-6t+13$ where $t$ is measured in seconds and the position is measured in meters. Find the position of the object when the velocity is 0. I'm confused on ...
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1answer
44 views

one sided derivatives

Show that if $ f'(a^+) $ and $f'_+(a) $ exist, then $ f'(a+) = f'_+(a) $. Here $ f'(a+) = \lim_{x \to a^+} f'(x) $ and $ f'_+(a) = \lim_{x \to a^+ } \frac{f(x) - f(a)}{x-a} $ $\textbf{Attempt:}$ ...
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2answers
60 views

finding function with limit and derivative

The limit is: $\lim_{x\to3}\frac{7^{3x}-7^9}{x-3}$ This is equal to the derivative of a function f at a point a (that is, f′(a)). how do you figure out what the function f is and the value of a ? I ...
1
vote
4answers
253 views

show $\sin(x)$ and $\tan(x)$ are increasing

Show that the functions sin and tan are each increasing on $(-π/2, π/2)$. Hence define the functions $\sin^{-1}$ and $\tan^{-1}$ (on $(-1,1)$ and $\Bbb R$ respectively), prove them differentiable, and ...
3
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3answers
112 views

Why is the power rule for derivatives not valid here?

I am stuck on an exercise where I have to figure out the derivative of $y = \frac{\sqrt{20-x^2}}{4}$. I realize that this equation can be rewritten as: $1/4 * \sqrt{20-x^2}$, so when I factor out the ...
1
vote
2answers
82 views

Derivative of $\frac{d}{dx}\{\exp \frac{-(x-\mu)^2)}{2 \sigma^2}\}$

How to find derivative of this complex exponent? $$\dfrac{d}{dx}\{\exp \dfrac{-(x-\mu)^2)}{2 \sigma^2}\}$$
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1answer
25 views

Is it permissible to locate the abscissa of extreme points of $y=f(x)$ by powering the function first for the sake of simplicity?

Let's take a simple example as follows, $$R=\sqrt{A^2+B^2 +2AB\cos \theta}$$ It represents the magnitude relation of vectors $\vec A$, $\vec B$, and $\vec R$ which is $\vec A +\vec B$. And we have ...
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votes
2answers
53 views

Unable to figure out how to solve for a final value

We want to minimize the equation below with respect to r. $$\frac{b}{r}(n+2^r)$$ where b is a constant. The professor suggested we take the derivative of the equation, set it equal to 0, and then ...
2
votes
5answers
107 views

Derivatives, when to use the chain rule, and when to use the formula.

When should I use the formula below, and when should I use the chain rule? Or does it not matter? I find using chain rule to be much faster and easier to solve. $$\lim_{x \to 0} ...
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1answer
55 views

What am I doing wrong? Deriving

I'm doing the same problem as the one on pg. 5 of this link: http://www.math.lsa.umich.edu/~pwn/01~ch1_solutions.pdf I got the same answer for my equation, although my approach was slightly ...
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1answer
938 views

How to find average rate change

Let $f(x)=3 \cdot x^2+5 \cdot x-4$ The average rate of change of f between x= 1 and x= 1.17 equals ? The instantaneous rate of change of f at x= 1 equals ? How do I do these ...
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1answer
37 views

$\sup_{x>0}\sqrt{\frac{2}{\pi}}\exp(x-\frac{x^2}{2})=?$

$$\sup_{x>0}\sqrt{\frac{2}{\pi}}\exp(x-\frac{x^2}{2})=?$$ I tried in the following way: $$\sup_{x>0}\sqrt{\frac{2}{\pi}}\exp(x-\frac{x^2}{2})$$ ...
3
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2answers
99 views

Prove the Fundamental Theorem of Calculus

Prove the Fundamental Theorem of Calculus with this hypothesis: If $f$ is integrable over $[a,b]$, if $g:[a,b]\rightarrow\Bbb R$ given by $g(x)=\int_{a}^{x}f(t)dt$ and $f$ is continuous in $x_0 \in ...
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1answer
312 views

What is the derivative of this trig function?

What is the derivative of $y=3x-5\cos^2(\pi x)$ ? What rule do I start with?
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votes
4answers
2k views

Find all points where the tangent line has slope 1.

Let $f(x)=x-\cos(x)$. Find all points on the graph of $y=f(x)$ where the tangent line has slope 1. (In each answer $n$ varies among all integers). So far I've used the Sum derivative rule for which ...
2
votes
5answers
101 views

Find $\lim_{x\to 0} \frac{\tan16x}{\sin2x}$

Find $\lim_{x\to 0} \frac{\tan16x}{\sin2x}$ I'm a little confused on limit trig. Am i suppose to simplify tan or do I use the derivative quotient rule? Please Help!!!
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1answer
119 views

Prove (local) converse to the implicit function theorem

The implicit function theorem tells us that: Given a level set $M^k = F^{-1}(F(p_0))$ of a smooth function $$F: \mathbb{R}^n \to \mathbb{R}^{n-k},$$ where $\operatorname{rk}{(Df)(p)} = n-k$ for ...
2
votes
2answers
533 views

The local lipschitz condition implies differentiability?

I know differentiability implies the local lipschitz condition. however, I am not sure the converse. Actually, I think it might be. The definition of the local lipschitz condition is that for $$ ...
2
votes
3answers
92 views

Using both Leibniz' notation and prime-notation for a derivative

I am presented with the following task: "Assume that the function $f(x)$ has the derivative $f'(x) = \frac{1}{x}$ and that $f$ is one-to-one. If $y = f^{-1}(x)$, show that $\frac{dy}{dx} = 1$. The ...
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0answers
43 views

How $k^∗$ be infinite and $\mu= \frac{\lambda}{m}$?

$$k^∗ = \sup_{x>0}\frac{\lambda^m x^{(m−1)}e^{(μ−λ)x}}{μΓ(m)}$$ (1) How will $k^∗$ be infinite if $m < 1$ or $λ ≤ μ$ ? taking the derivative of the right-hand side above and set it to zero, ...
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0answers
35 views

How to prove that the following derivative is negative?

I was wondering how to prove the following partial derivative is negative: \begin{eqnarray} \frac{\partial }{\partial \alpha} \int\limits_{-\bar{x}}^{\bar{x}} (n-1)F^{n-2}_\epsilon(\frac{x}{\alpha}) ...
2
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3answers
52 views

Evaluating the following expression

Why is the value of the following expression equal to 0? I have a feeling that I need to apply L'Hopital's rule, but I do not know where. $[-x(1-F_X(x)]\Big|_0^{\infty}$, where $F_X(x)$ is the ...
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2answers
67 views

Evaluating $xe^{-x/\lambda}\big|_0^\infty$ with and without L'Hopital's Rule

How to evaluate $$\left. \frac x{e^{\frac x\lambda}} \right|_0^\infty$$ using: L'Hopital's Rule Without using L'Hopital's Rule? Or should I use the rule partially for $\infty$ and not use it for ...
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1answer
35 views

Calculating limit using L'Hopital's Rule

I need to compute the expression $xe^{-x/a}$ with the limits of $x$ from infinity to $0$. When I use L'Hopital's Rule, however, I do not get the correct answer of $0$. What is the problem?
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1answer
82 views

Why does L'Hopital's rule give the wrong answer?

I have this function $\frac{\sin^2 x}{1-\cos x}$. $\frac{\sin^2x}{1-\cos x}=\frac{1-\cos^2x}{1-\cos x}=1+\cos x\;$. Thus the derivative of $1 + \cos x\; = -\sin x\;$. However by, L'Hopital's rule, ...
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0answers
56 views

Abbreviation for $n$ times differentiable, with $n$th derivative bounded?

Are there convenient abbreviations in use for the following sets? The set of functions which are $n$ times differentiable, with first $n-1$ derivatives continuous (obviously the last part is ...
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votes
2answers
163 views

Does differentiation symbol need parentheses or?

Suppose I have this expression: $$\frac{d}{dx}(e^{x})^2 + 6$$ Does it mean to differentiate $6$ too or just the first term? This is an exercise on a calculus course that I'm doing on Coursera. ...
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1answer
39 views

Partial Differentiation.

$$\frac{\partial}{\partial y}(y-z)^{\frac{1}{2}}=???$$ I can differentiate it $$\frac{d}{d y}(y-z)^{\frac{1}{2}}=\frac{1}{2}(y-z)^{\frac{-1}{2}}$$ But i don't know when it is Partial.
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1answer
216 views

what does derivative do in basis of function

I would like to understand how the derivative affects a basis of a vector space of polynomials. For instance suppose we have the collection of quadratic polynomials with real coefficients which has ...
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4answers
175 views

Derivative of $f(x)=(3x+4)(x-2)^3$

I have the following function: $$ f(x)=(3x+4)(x-2)^3$$ Now I want the derivative. The book says that it would be: $$f'(x)=(12x+6)(x-2)^2$$ I just don't understand how they got there. Can anyone ...
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1answer
141 views

Gradient of a scalar function with respect to a matrix

I need to calculate $\dfrac{\partial}{\partial K}f(K)$, with: $$ f(K)=-\frac{1}{2}(u-Kx)^T\Sigma^{-1}(u-Kx)$$ $K$ and $\Sigma$ are $n\times n$ matrices, $\Sigma$ is symmetric, $u$ and $x$ are column ...
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0answers
169 views

Text with alternative definition of “derivative”?

Instantaneous rates of change are conventionally defined as limits of difference quotients. Rates of things moving at constant speed are definable without delicate issues. If I pass someone moving ...
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1answer
59 views

Did I write down the derivative product rule correctly for $g(x)=(f(x))^2$

Suppose that $f(4)=5$ and $f'(4)=5$ . Use the product rule to determine the value of $g'(4)$ where $g(x)=(f(x))^2$ So I'm writing this problem as: ...
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1answer
43 views

$f(4)=3$ and $f'(4)=7$. If $g(x)=x^2f(x)$ then find $g'(4)$.

Suppose that $f(4)=3$ and $f'(4)=7$. If $g(x)=x^2f(x)$ then find $g'(4)$. I'm a little confused on this problem. If anybody can help me with this problem I would appreciate it.
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1answer
35 views

Difficulties in the operator notation in partial differentiation

While trying partial differentiation, I came to a dead end, where the book didn't provide me a satisfactory explanation. First of all, what does the notation mean: Say I have a relation $$x^2{\partial ...
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1answer
30 views

Rescaling Dataset by a constant value

I have two sets of measurements that measure one thing. I would like to graph the relative heights of the measurements, however one set is nearly consistently higher than the other one. I would like ...
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0answers
12 views

If $d/dx_t ({\dot y}_t/y_t) > 0$ and $dy_t/dx_t < 0$ what can I then say about the sign of $d{\dot y}_t/dx_t$?

Assume that the rate of change in $y_t$ over time is ${{{{\dot y}_t}} \over {{y_t}}} = {x_t}$, where $x_t >0$. The derivative of this expression with respect to $x_t$ will be positive (well, it ...
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1answer
102 views

Differentiability of arctan function

I should evaluate in which areas/intervals this function is differentiable and then differentiate. $$ \arctan\left({\sqrt{\frac{x+1}{x-1}}}\right) $$ So my approach would be: assume continuity and ...
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1answer
42 views

A basic doubt on derivatives

I have one question regarding differentiation : 1) Why in the definition of Taylor's series it requires the function to be "continuously" differentiable $m$ times in $[a,b]$? The book I am following ...
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0answers
27 views

A doubt on some algebraic manipulation of a multivariable differentiation formula

In $\Bbb R^n$ we have a map $f:\Bbb R^n \to \Bbb R$ . Now in a book the following is written in some derivation : ...
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3answers
338 views

Finding the tangent line to a curve

Find an equation for the tangent line to the curve $$x\sin(xy-y^2)=x^2-1$$ through the point $(1,1)$.
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1answer
34 views

Equations of a moving particle on a plane

Equations of a moving particle on a plane: $$\mathbf{r}(t)=x(t)\mathbf{i} + y(t)\mathbf{j}$$ $$\mathbf{v}(t)=\dot{\mathbf{r}}=\dot{x}(t)\mathbf{i} + \dot{y}(t)\mathbf{j}$$ ...