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

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1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
0
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0answers
26 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
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2answers
58 views

Finding the Derivative of $\sqrt{x}$

How can I find the derivative of $\sqrt{x}$ using first principle. Specifically I'm having difficulty expanding $\sqrt{x + h}$ or rather $(x + h)^.5$. Is there any generalized formula for the ...
0
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1answer
37 views

Local extremes of $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$

The task is to find local extremes of $f: \mathbb R \to \mathbb R$, $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$ There is theorem that if $x_{0}$ is local extreme of $f(x)$ then $f'(x_0) = 0$ So ...
1
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1answer
12 views

Proof that a derivative's points of discontinuity are all essential

I'm reading Wikipedia's article on Darboux's theorem, and it says the following: "Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the ...
2
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6answers
74 views

Derivation for the derivative of $a^{t}$ from The Equation

In Calculus, the Equation is known as: $$f'(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ This equation allow us to find the derivatives of functions. Let's try this with the exponential ...
-3
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0answers
27 views

Use the alternative form of the derivative to find the derivative at $x=c$ [closed]

$f(x)= x^2-1,$ at $c=2$. $g(x)= \sqrt{|x|},$ at $c=0$. $h(x)= |x-5|,$ at $c=-5$.
0
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1answer
40 views

Value of the derivative of a function at a point depends only on the germ at that point

Suppose that f : I → R is a $C^∞$ function defined on an open subset I ⊆ R. How can I show that for $a \in I$ the value $f^n (s)$, n = 1, 2, 3, . . . of the derivative of $f$ of order n at s depends ...
4
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3answers
75 views

Implicit Differentiation. Please help me understand why!

I am trying to understand implicit differentiation; I understand what to do (that is no problem), but why I do it is another story. For example: $$3y^2=5x^3 $$ I understand that, if I take the ...
1
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1answer
66 views

-relationship between a function and a tangent line

$f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function at $x=a$. Show that $f$ has derivate at $x=a$ iff there's only a $L(x) = m(x-a)+b $ such that $$ \lim_{x \to a}\frac{f(x)-L(x)}{x-a} = 0 ...
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2answers
51 views

$n$-th derivative of $(ax+b)^{-m}$ [closed]

How to find the $n$-th derivative of $(ax+b)^{-m}$ ?
3
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1answer
58 views

Example of a function in $L^2(\mathbb{R})$ with derivative not in $L^2(\mathbb{R})$.

We know examples of a function which doesn't lie in $L^2(\mathbb{R})$ with derivatives in $L^2(\mathbb{R})$: $$f_1(x) = \mathrm{arctg}(x) \notin L^2(\mathbb{R}), \qquad f_1'(x) = \frac{1}{x^2+1}\in ...
1
vote
1answer
43 views

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+…+na_n|\leq1$

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+\cdots+na_n|\leq1$ When we try to differentiate it on both sides wrt $x$,then modulus ...
2
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2answers
82 views

Maxima/Minima of absolute function

Given $a_i=\{a_1,\dots,a_n\}$ and function $$f(x)=\sum_{i=1}^n{|x-a_i|}^3$$ I need to find minimum value of $f(x)$. As far my understanding goes the derivative is given by: $$f'(x) = ...
1
vote
4answers
57 views

If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$

If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$ I tried to solve it.But i got stuck after some steps. $x^4+7x^2y^2+9y^4=24xy^3$ ...
1
vote
1answer
29 views

$y=\tan^{-1}\frac{1}{x^2+x+1}+\tan^{-1}\frac{1}{x^2+3x+3}+\tan^{-1}\frac{1}{x^2+5x+7}+\tan^{-1}\frac{1}{x^2+7x+13}…$to n terms.

Prove that if $y=\tan^{-1}\frac{1}{x^2+x+1}+\tan^{-1}\frac{1}{x^2+3x+3}+\tan^{-1}\frac{1}{x^2+5x+7}+\tan^{-1}\frac{1}{x^2+7x+13}......$to n terms.Then ...
0
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3answers
29 views

Solve the following differential equation of one independent variable

I want to solve the following differential equation $$\frac{dy}{dx}=e^{x-y}(e^x-e^y)$$ I am trying to separate $x$ and $y$ in this way : $\frac{dy}{dx}=e^{x-y}(e^x-e^y)=e^x(e^{x-y}-1)$ Put $x-y=z$. ...
1
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3answers
41 views

If $f’(x) = \sin x + (\sin4x)(\cos x)$, then $f’(2x^2 + \pi/2) $is?

If $$f'(x) = \sin x + \sin4x \cdot \cos x,$$ then $$f'(2x^2 + \pi/2)$$ is? Given answer: $$4x\cos(2x^2) – 4x\sin(8x^2) \sin(2x^2)$$ I tried and I'm getting the answer as $\cos(2x^2) - ...
3
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1answer
35 views

Does there exist a function that is differentiable everywhere with everywhere discontinuous partial derivatives?

Does there exist a function that is differentiable everywhere with everywhere discontinuous partial derivatives? I just had this doubt, talking about first order partials.
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2answers
46 views

Laplacian of $|f|^p,$ where $f$ is holomorphic

I have to prove that if $f$ is a homolorphic function that doesn´t vanish on its domain then $\triangle |f|^p=p^2 |f|^{p-2} |\frac{\partial f}{\partial z}|^2$ . My attempt: I take $|f|^p=(z ...
3
votes
2answers
58 views

If $f(x) = \cos x\cos2x\cos4x\cos8x\cos16x$, then $f’(\pi/4)= ?$

If $f(x) = \cos x\cos2x\cos4x\cos8x\cos16x$, then $f’(\pi/4)= ?$ Ans: $\sqrt{2}$
1
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4answers
33 views

Finding the slope of a curve with a given point [closed]

I am just stuck and cannot see how to solve this question, I've have a complete mind blank. Find the slope of the curve $$y= 2x^3 − 8x^2+1$$ at the point $(2, -15).$
7
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2answers
454 views

Calculus Paradox. I mean, what's wrong with what I think? [closed]

Is not calculus based on the paradox that the closest point to a point A is a distinct point B which is the point A itself? For example, if we consider the limit, $$ \lim_{x\to2} \frac{x^2-4}{x-2} ...
0
votes
1answer
13 views

Tangent from points on a curve meeting the curve again and again

tangent at a point C1 on the curve y=x^3 meets the curve again at C2 .the tangent at C2 meets the curve at C3, and soo on, so that the abscissa of c1,c2,c3.....,Cn form a G.P. find the ratio of area ...
1
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1answer
37 views

Solve this question involving temperatures?

So I am given 2 formulas: $$ \frac{dT}{dt}=-k(T_t-T_s)$$ Where $\frac{dT}{dt}$ rate at which the object's temperature is changing $T(t)$ is the temperature of the object at time $t$ $T(s)$ is the ...
1
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2answers
35 views

Check Differentiability

chech whether the function is differentiable at $x=0$ $$f(x)=\left\lbrace \begin{array}{cl} \arctan\frac{1}{\left | x \right |}, & x\neq 0 \\ \frac{\pi}{2}, & x=0\\ \end{array}\right.$$ ...
0
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1answer
32 views

How can $ (D^2 +1)y $ be solved such that it's equal to $x \cos x$?

Can anyone provide solution for $(D^2 +1)y$ such that it's equal to $ x \cos x$ or vice versa?
2
votes
1answer
53 views

Computing the integral of $-1/f''$

I think this is a very silly question but I have some problems nonetheless. If I know that $g'=-\frac{1}{f''}$, is then $$ g=(f')^{-1}? $$
1
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1answer
38 views

Derivatives and the cotangent space

In Differentiable Manifolds, the derivative of a function $f: M \rightarrow \mathbb{R}$ at $a$ denoted by $(df)_a$ is defined as its image in the cotangent space: $T_a^* = C^\infty(M)/Z_a$, where ...
3
votes
1answer
68 views

Prove the Lipschitz constant must be less than 1.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
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0answers
39 views

Finding a smooth function where $f^{(n)}(0)=(n-2)!$

Is there a function that is $C^\infty \big((-1,1) \big)$ function where $f^{(n)} (0)=(n-2)!$ for every $n \ge 2$? If there is such a function find its formula expressed in terms of elementary ...
1
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1answer
17 views

Newton Raphson, given derivatives

I'm trying to calculate the value of $f(x_1)$ with Newton Raphson's method. The following information is given: $x_0=0$ $x_1=1$ $f(x_0)=2$ $f'(x_0)=0$ $f'(x_1)=0$ $f''(x_0)=0$ $f''(x_1)=0$ ...
3
votes
1answer
126 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
1
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1answer
37 views

The behavior of $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ near $0$, where $\alpha \ge 1$.

Consider $\alpha \ge 1$. Let $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ and let $f(0)=0.$ In order to find the sign of $f'(x)$ when $\alpha \ge 1$ it is necessary to decide if ...
0
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1answer
16 views

Definition of continuously differentiable for functions of several variables

When we say that a function $f:\mathbb{R}^m\to\mathbb{R}^m$ is $C^1$, what exactly does this mean? Does it mean that all the directional derivatives are continuous individually (I am sure not), or ...
0
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1answer
39 views

How do I show this function is monotonically decreasing?

Let $p = $Probability of head on a coin toss ; $p < 0.5$ (biased coin). $f(k) =$ Probability that heads is the majority from $k$ tosses, where $k$ takes odd values. I want to show that $f(k)$ ...
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2answers
35 views

Linear Algebra Change of Basis problem

So, $\mathbb{P}_2$ is the vector space of all polynomials with degree less than or equal to 2 and that $E=\{1,t,t^2\}$ is a basis for $\mathbb{P}_2$ We define $p_1(t)=1+2t$ $p_2(t)=t-t^2$ ...
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1answer
66 views

How can I prove $x^3\, \frac{d^3 y}{dx^3} = \Delta(\Delta-1)(\Delta-2)y$?

This equation is used to solve Cauchy Euler Equation As it can be seen author has provided explanation of the fact how ...
4
votes
1answer
94 views

Find a formula for $f''$ in terms of $f$, where $f\gt 0$ and $(f')^2=f-\frac{1}{f^2}.$

Problem: Suppose that a function $f \gt 0$ has the property $$ (f')^2=f-\frac{1}{f^2} $$ Find a formula for $f''$ in terms of $f$. Hint: Use Theorem 7. Theorem 7: Suppose that $f$ is ...
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0answers
17 views

Computing $\bigtriangledown r^m$ knowing the vector $\boldsymbol r$

I am asked to compute $\bigtriangledown \cdot \boldsymbol r$ and $\bigtriangledown r^m$ for $m$ constant, where $\boldsymbol r =x \boldsymbol i+ y\boldsymbol j +z\boldsymbol k$ and $r= |\boldsymbol ...
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votes
1answer
22 views

$D\det_A$ exists and equals $D\det_A (H)=\det (A) \operatorname{trace} (A^{-1}H) $? [closed]

Consider the determinant function $\det : M_n(\mathbb R ) \to \mathbb R$ , the is it true that $D\det _{A}$ exists ? Does it exist if $A$ is assumed to be invertible also and at $H \in M_n(\mathbb R)$ ...
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1answer
54 views

Prove there exists a unique local inverse.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
0
votes
1answer
56 views

Prove the following function is Lipschitz with constant less than 1.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
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votes
1answer
44 views

Find partial derivatives, given directional derivatives.

You are given that the directional derivatives of a function $f$, at the point $(a, b)$, in the direction of the two vectors $(1, 2)$ and $(−1, 1)$, are $2$ and $3$ respectively. Find the partial ...
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1answer
29 views

Varying definitions for concavity of a function

I've been working on concavity of functions and have noticed that different texts define this notion in different ways. Specifically, some texts include the endpoints of an interval when describing ...
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vote
2answers
38 views

Polynomial must be monotone between its extrema

Suppose that the polynomial function $f(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ has $k_1$ local maximum points and $k_2$ local minimum points. Show that $k_2=k_1+1$ if $n$ is even, and $k_2=k_1$ if $n$ is ...
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vote
2answers
51 views

Maximizing area under $y=e^{−{∣x∣}}$

The coordinates of the point $M(x,y)$ on $y=e^{−{∣x∣}}$ so that the area formed by the coordinates axes and the tangent at $M$ is greatest is what? I tried to plot the graph but after that I'm not ...
1
vote
0answers
19 views

Calculation of a Frechet derivative

Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real ...
0
votes
0answers
50 views

Why doesn't $\dot{x}=-\dot{z}sin(φt)-φzcos(φt)+\dot{y}cos (φt)-φy sin(φt)$

\begin{align} \ {x}=y cos (φt) -zsin(φt) \end{align} \begin{align} \dot{x}=\dot{y}cos (φt)+(-φy sin(φt))-(\dot {z}sin(φt)+φzcos(φt)) \end{align} \begin{align} \dot{x}=\dot{y}cos (φt)-φy sin(φt)-\dot ...
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0answers
15 views

Question related to differentiation of linear and quadratic functions [duplicate]

Sq(1)=1 Sq(2)=2+2 Sq(3)=3+3+3 Sq(4)=4+4+4+4 Like this Sq(x)=x+x+x+x+--------+x Differentiating both sides d/dx[sq(x)]=d/dx(x+x+x+x+--------+x) This implies 2x=1+1+1+1+----+ x times 2x=x How it is ...