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

learn more… | top users | synonyms (2)

0
votes
0answers
13 views

General case for differentiation under the integral sign

What is the most convenient way to decide if we can differentiate under the integral sign? If the integrant is a smooth function, could we do so?
0
votes
1answer
33 views

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value, say $F(a)$, and that $F(x)$ has a minimum value $e^{-e/2}$

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value say $F(a)$,and that $F(x)$ has a minimum value $e^{-e/2}$. I differentiated the function $f(x)=e^{-ax}x^{a^2}$ to get ...
1
vote
3answers
37 views

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$.find the value of $a_2+11a_3+70a_4$ I differentiated ...
1
vote
1answer
43 views

Proof of Green's identity

Can anyone explain to me how to prove Green's identity by integrating the divergence theorem? I don't understand how divergence, total derivative, and Laplace are related to each other. Why is this ...
0
votes
1answer
32 views

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$?

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$? Let $a,a,b$ are the sides of the isosceles triangle whose perimeter is ...
0
votes
1answer
38 views

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length $l$ of the median drawn to its lateral side. I tried to solve this ...
1
vote
1answer
23 views

Find the values of the derivatives of the integral with a variable inside its limits.

$\require{cancel}$ Problem: I have the function $g: \mathbb{R} \to \mathbb{R}$ defined as $$ g(x)=\int^{(1+x^2)}_{-(1+x^2)} sin(t^3)\ dt,\ x \in \mathbb{R} $$ I would like to calculate values of ...
0
votes
1answer
28 views

If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$

I wanna prove this problem. I tried it with Mean Value Theorem but cannot proceed to any plausible result. So could I have some hints?
0
votes
0answers
27 views

Find the maximum volume of the cylinder.

A cylinder is obtained by revolving a rectangle about the $x-$axis,the base of the rectangle lying on the $x-$axis and the entire rectangle lying in the region between the curve $y=\frac{x}{x^2+1}$ ...
1
vote
0answers
39 views

Questioning the differentiability of $f(x,y)$

$$f(x,y)=\begin{cases} y- \frac{e^{x^2+y^2}-x^2-y^2}{x^2+y^2},& x^2+y^2 \neq 0. \\ -1, & x=y=0 \end{cases}$$ I keep runnung into trouble with these types of questions. The way I do them is ...
1
vote
2answers
40 views

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, then $p=q$.

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, defined by: $$ F = \begin{cases} \hfill q \hfill & \text{X $\geq$ a} \\ \hfill p \hfill & ...
0
votes
3answers
33 views

Prove that the following functions is differentiable on $(-1,1) \times \mathbb R$

$$f(x,y)=\begin{cases} \frac{\tan x}{x}+y, & 0<|x|<1 \\ 1+y,& x=0 \\ \end{cases}$$ Prove that it is differentiable on $(-1,1) \times \mathbb R$. I use the Frechet definition of ...
0
votes
2answers
35 views

Differentiation using Chain Rule

Find $\frac{dy}{dx}$ if $y=7+5^{x^2+2x-1}$. So far I have done $\frac{dy}{dx}=(5^{x^2+2x-1})'$. Now, the RHS can be found by $(e^{\ln 5\cdot (x^2+2x-1)})'=e^{\ln 5\cdot (x^2+2x-1)}(x^2+2x-1)'\ln ...
0
votes
0answers
21 views

Computing Partial Derivatives (basic)

When I am asked to compute a partial derivative of $f_x$ for $f(x, y)=x \ln(xy)$, I treat this the same as $\frac{d}{dx} (x \ln(xy))$ which I then just simply apply the chain rule and get $\ln (xy) ...
0
votes
0answers
24 views

How to differentiate $\sum_{t=0}^{T}$ problem?

Define $F(C, H)=\sum_{t=0}^{T}\beta^t (logC_t+\mu log H_t)$ where $\beta$ and $\mu$ are constant. I expand this function like $F(C, H)=\beta^0 (logC_0+\mu logH_0)+ \beta^1(logC_1+\mu logH_1)+\dots$ ...
-3
votes
1answer
56 views

why the integral of $\frac{dy}{y} =\ln(y)$?

I mean if I differentiate $\ln(y)$ the result will be $\frac{dy}{y}$ ? . What I know the diffential of $\ln(x)$ = $\frac{1}{x}$ right?. And following this idea what is going to happen if we ...
1
vote
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
votes
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 ...
0
votes
2answers
60 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
votes
1answer
38 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
vote
1answer
13 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
votes
6answers
75 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 ...
0
votes
1answer
41 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
votes
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
vote
1answer
67 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 ...
-2
votes
2answers
51 views

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

How to find the $n$-th derivative of $(ax+b)^{-m}$ ?
3
votes
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
votes
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
30 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
votes
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
vote
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
votes
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.
1
vote
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
vote
4answers
34 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
votes
2answers
458 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
vote
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
vote
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
votes
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
vote
1answer
40 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
69 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 ...
-3
votes
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
vote
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
128 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
vote
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
votes
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 ...