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

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4answers
116 views

The Derivative of a Derivative?

For homework, one of the critical questions asks, "Is it possible to find the derivative of a derivative, why or why not? Provide a proof in your explanation." My first thought was that you could do ...
3
votes
5answers
122 views

Chain Rule: Is the notation $\frac{dy}{du} \cdot \frac{du}{dx} = \frac{dy}{dx}$ accurate?

My question is if it is okay / mathematically rigorous to write the Chain Rule like that (the Leibniz way). I thought that $dx$, etc. do not follow the rules of algebra and cannot be treated as such. ...
2
votes
2answers
225 views

Show that if $f(x)= \sum\limits_{i=0}^n a_i x^i$ and $a_0+\frac{a_1}{2}+\ldots+\frac{a_n}{n+1}=0$, then there is an $x \in (0,1)$ with $f(x)=0$

Show that if $f(x)= \sum\limits_{i=0}^n a_i x^i$ and $a_0+\dfrac{a_1}{2}+\ldots+\dfrac{a_n}{n+1}=0$, then there is an $x \in (0,1)$ with $f(x)=0.$
0
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1answer
53 views

Show that if $f''(x)=-k^2f(x)$, then $f(x)= A \sin (kx) + B \cos (kx)$ [closed]

Show that if $f''(x)=-k^2 f(x)$, then $f(x)= A \sin (kx) + B \cos (kx)$
0
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3answers
136 views

Linear Programming and differentiation, why can't we differentiate to find the optimum solution?

I do understand that differentiating a linear function (for a maximization) subject to some linear restriction (such as the problem $p=ax+by$ s.t. $cx+dy \leq m$) won't necessarily give me the right ...
1
vote
3answers
77 views

Prove that a piecewise function $f$ is differentiable at $0$

$f(x)= e^{-\frac{1}{x}}$, $x>0$ $f(x)=-x^2$, $x\leq0$ This is a function that is defined for $x$ here above . How can I prove that the derivative of $f$ exists at $x=0$.
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0answers
19 views

How do you know when the derivative is a matrix or a sum?

For the derivative of a function, sometimes I see the derivative written as a matrix of the partial derivatives, and sometimes I see the derivative written as a sum of the partial derivatives. When do ...
3
votes
3answers
63 views

calculus question attempt

find the maximum value of the function $$y = 15 \sin x -8 \cos x $$ attempt at a solution: deriving: $y' = 15\cos x +8\sin x $ equating to zero and doubling by $ 1/\cos x$ (Im not sure this is ...
2
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0answers
36 views

How does the third derivative visually affect its function? [duplicate]

I find the lack of information on the internet about the third derivative astounding. When the first derivative of a function touches or crosses the x-axis, it is a critical point and possibly an ...
0
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1answer
51 views

Find equation of tangents to curve

Find the equations of tangents to the curve $3x^2 - y^2 = 8$ which pass through the point $(4/3,0)$ I googled it and got this yahoo answers question which has 2 contradicting answers. I did ...
0
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2answers
110 views

Find the mistake.

We all know the product rule of differentiation That, $(uv)'=u'v+v'u$ I tried proving the same and what I have done I'm showing you, you have to find the mistake in my proof (if you are interested ) ...
1
vote
4answers
399 views

Calculus Derivative - Finding unknown constants

Determine the constants $a$, $b$, $c$, and $d$ so that the curve defined by $y = ax^3 + bx^2 + cx + d$ has a local maximum at the point $(2, 4)$ and a point of inflection at the origin. Sketch the ...
0
votes
1answer
23 views

Non-linear estimate parameter

I have one non-linear function that define $$E_x(a,b)=\int K_\sigma(y-x) \cdot(b-b. e^{-a\cdot f(y)} \,) dy$$ where $y$ is neighboor points of $x$; $f(y)$ is a function of $y$; and $a$ is constant. ...
1
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1answer
1k views

$x$-coordinates of points where tangent line is horizontal or vertical. Using implicit differentiation

For the implicit equation $x^3 + y^3 - xy^2 = 8$ Determine the exact x-coordinates of all points where the tangent line is horizontal or vertical I figured out $\dfrac{dy}{dx} =\dfrac{y^2 - 3x^2}{ ...
0
votes
2answers
65 views

Finding the missing values of a function

If the graph of the function $g(x) = \frac{ax+b}{(x-1)(x-4)}$ has a horizontal tangent at $(2,-1)$, determine the values of $a$ and $b$. I am unsure on how to go about solving this question. I found ...
0
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1answer
31 views

derivative of the exponential integral

I have one function that is defined: $$E_x(f)=\int K_\sigma(y-x) \cdot e^{-a\cdot f(y)} \, dy$$ where $y$ is neighboor points of $x$; $f(y)$ is a function of $y$; and $a$ is constant. I want to ...
1
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2answers
91 views

twice differentiable function [closed]

Is this function twice continuously differentiable? $$ f(x) = \sum_{i=1}^{n}\big(\max\{0,a_i-x_i\}\big)^2 $$ where $(x,a\in\mathbb{R}_+^{n})$. Any hints?
0
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1answer
64 views

Trouble reading directional derivative proof

I'm reading Vector Calculus from http://mecmath.net/. This is a free PDF book for students of Calculus III. In section 2.4 (page 78) it introduces the directional derivative and theorem 2.2: So far ...
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2answers
110 views

How can I solve this question?

Compute the value of the following improper integral. If it is divergent, type "Diverges" or "D". $$\int_0^2 \frac{dx}{\sqrt{4-x^2}}$$ Do I make $u= 4-x^2$ then $du= -2x \, dx$ Not exactly sure..
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2answers
72 views

Multivariable calculus - explain what the teacher did

The teacher gave this exercise: Find $D_f(a)$ when $f: \mathbb R^n \to \mathbb R$, $f(x)=<x,\xi>^2$ where $\xi \in \mathbb R^n$. What I did: I wrote it as $$f(x)= (\sum_{i=1}^{n}x_i ...
0
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0answers
38 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
2
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2answers
67 views

Find constants $a$ and $b$ such that $ \lim_{x \to 0} \frac{1}{bx-\sin x} \int^x_0 \frac{t^2dt}{ \sqrt{a+t}}=1$

Find constants $a$ and $b$ such that $$ \lim_{x \to 0} \frac{1}{bx-\sin x} \int^x_0 \frac{t^2dt}{ \sqrt{a+t}}=1$$ First,$a$ should be positive to make sure the limit is meaningful as $x \to 0^-$ ...
0
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1answer
52 views

can $f$ twice differentiable on $(0, 1)$ and continous on $[0,1]$ have a derivative discontinuous on $[0,1]$

As I wrote in the title can the first derivative of $f$ be discontinous on $[0,1]$ given $f(x): [0,1] \rightarrow \mathcal{R}$ twice differentiable on $(0,1)$ and continuous on $[0,1]$? My idea is to ...
2
votes
2answers
53 views

evaluate $\lim_{x \to 0+} \frac{x-\sin x}{(x \sin x)^{3/2}}$

Evaluate $\lim_{x \to 0+} \frac{x-\sin x}{(x \sin x)^{3/2}}$ This is an exercise after introducing L'Hopital's rule. I directly apply L'Hopital's rule three times and it becomes more and more ...
0
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2answers
31 views

Growth of Derivative

Assume that $f \in C^1(\mathbb{R})$ such that $\lim_{x \rightarrow 0} x^{-(k+1)} f(x) = 0$ for some $k \in \mathbb{N} = \{1, 2, 3, \ldots \}$. Do we have $\lim_{x \rightarrow 0} x^{-k} f'(x) = 0$? I ...
2
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5answers
97 views

Derivative of $ h(t)= \sin (\cos^{-1}t$)?

Can someone please explain the steps/rules I need to preform to find the derivative of $h(t)= \sin (\cos^{-1}t)$? I tried to used the product rule, and realized it was obviously a failure. Using ...
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0answers
40 views

prove that if f is $C^1$ (meaning the derivative $f´$ is continuous) then it can be represented as the sum of an increasing and a decreasing function

prove that if f is $C^1$ (meaning the derivative $f´$ is continuous) then it can be represented as the sum of an increasing and a decreasing function I can´t find any solution for this problem ...
0
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1answer
68 views

Critical points and absolute extreme values on given interval

I'm back with a question! I am working on a homework problem and I got stuck. I'm asked to 1.find the critical points of f on the given interval 2. fine the absolute extreme values of f on the given ...
1
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1answer
136 views

Strictly monotone real function: stationary point, non-differentiable point

If we have a real function $f$ that is strictly increasing (or strictly decreasing), what can we say about measure and cardinality of stationary points/points with no derivative. In particular: -Is ...
0
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1answer
36 views

Differential equation, symmetric about 0?

Solving the following numerically (with different values of $u(-1)$) $(2-\cos(\pi x))u''(t) + u(t) = 1$ and $u(-1) = u(1)$ the solutions seem to be symmetric about $0$. Is it true in general (ie no ...
0
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2answers
46 views

Find derivative of function

I need help in finding the derivative. I don't even know where to begin with it. I'm learning chain rule in school and do not see how I can apply that here. $$ f(x)=\left(\dfrac {x+1}{x^2+8}\right)^6 ...
0
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0answers
42 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
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2answers
41 views

Derivatives of Logarithmic function

Determine $f'(x)$ for $f(x) = ln(x + \sqrt{x^2 + 1})$ My handbook has the answer as $\displaystyle\frac{1}{\sqrt{x^2 + 1}}$ with no steps on how they got there. I tried to get there, but somewhere I ...
0
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1answer
48 views

if $f'''(x)$ is continuous everywhere and $\lim_{x \to 0}(1+x+ \frac{f(x)}{x})^{1/x}=e^3$ Compute $f''(0)$

if $f'''(x)$ is continuous everywhere and $$\lim_{x \to 0}(1+x+ \frac{f(x)}{x})^{1/x}=e^3$$ Compute $f''(0)$ The limit equals to $$\begin{align} \lim_{x \to 0} \frac{\log(1+x+ ...
1
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1answer
24 views

Partial derivative identity help

Please can some one give me the proof for this, ( I think y and x can be written as parametric equations in terms of C)(∂y/∂x) = (∂y/∂c) / (∂x/∂c))
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3answers
166 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
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0answers
48 views

Taylor series like polynomials

Let $U$ be an open subset of $R^n$ and $f:U\rightarrow \mathbb{R}$ a function and $x\in U$ such that in a small neighbourhood of $x$ and for $\epsilon \in \mathbb{R^b}$ sufficiently small we have the ...
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1answer
56 views

Does the little-oh relation remain if $f(x)$ and $g(x)$ both integrate or differentiate?

Give two functions $f$ and $g$ with derivatives in some interval containing 0,where $g$ is positive.Assume also $f(x)=o(g(x))$ as $x \to 0$. Prove or disprove each of the following statements: ...
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0answers
52 views

Covariant Derivatives of contravariant vector in curvilinear coordinates

$$D_mA^p = \partial_mA^p + \Gamma^p_{mn} A^n$$ so $$D_kD_mA^p = D_k(\partial_mA^p + \Gamma^p_{mn} A^n)$$ $$D_kD_mA^p = \partial_k(\partial_mA^p + \Gamma^p_{mn}A^n) + \Gamma^p_{kl}(\partial_mA^l + ...
2
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1answer
66 views

How can I complete this proof?

I have finished all but the last line. Thanks in advance for any help. Proof statement: If $X$ and $Y$ are Banach spaces and $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ are both differentiable, then ...
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1answer
113 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let two points $$z_1 = \frac{-1+i\sqrt3}{2}\quad\text{and}\quad z_2 =\frac{-1-i\sqrt3}{2}.$$ I am trying to show that there is no point $w$ on ...
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0answers
21 views

Is there anything wrong with this line of reasoning?

Proof statement: If $f:X \rightarrow Y$ is a bounded linear map, then $Df(x)=f$ for all $x \in X$, where $X$ and $Y$ are Banach spaces. Proof: Consider $$\lim_{h\rightarrow ...
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1answer
85 views

Bounded Jacobian implies uniform continuity

I am trying to solve the following problems but I am not sure what the difference between the 2 problems is. 1) Prove that is $U = B_r(x)$ (open ball centered at $x$ with radius $r>0$) is an open ...
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4answers
58 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
2
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1answer
100 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
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3answers
108 views

Why are there so many notations for differentiation?

There are so many notations for differentiation. Some of them are: $$ f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad D f(x)\qquad y^\prime\qquad D_x f(x) $$ Why ...
-1
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1answer
148 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
0
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1answer
53 views

Finding the derivative of analytic polynomials

I have just started studying complex analysis and i am stuck with one question. My book says, the derivative of an analytic polynomial with respect to $z$ is equal to the partial derivative of that ...
1
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1answer
81 views

$f$ continuous, monotone, what do we know about differentiability?

I am interested in knowing what we can say in general about when a continuous function $f:\mathbb{R} \to \mathbb{R}$ is differentiable. To my mind, there are various ways a continuous function can ...
0
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3answers
175 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...