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

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4
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1answer
43 views

Finding Solutions (with only pen and paper)

For what least value of $k$ does the equation:$$e^x=kx^2$$ Have 3 solutions? Let $f(x)=e^x$ and $g(x)=kx^2$. For a positive $k$, drawing a rough graph of $f(x)$ and $g(x)$ does show 2 solutions of ...
0
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1answer
25 views

Does infinitely differentiable imply analytic? [duplicate]

I know that analytic implies infinitely differentiable, but is the converse always true as well?
0
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1answer
27 views

Gradient of high dimensional function

I hope this is the right forum for this. Here are the givens: $X$ is a matrix $1230 x 30$ with the following properties: In a line, all values are $0$ except one $1$ and one $-1$. $Y = ...
0
votes
0answers
39 views

The most general way to prove differentiability over an interval

Preface: I'm going to try and make this question as general as I possibly can, as there are many different extensions of Calculus (Single-Variable, Multi-Variable, Vector, Tensor etc.), in which ...
1
vote
0answers
30 views

How do you call this formula in math terms? Quotient derivative?

Sorry for that noob question, but I've been searching for ages without finding something... I've got a series of values $x(n)$. Now to get the derivative I would subtract the current value from the ...
1
vote
2answers
25 views

Intuition about theorem relating continuity and differentiability

I am following Apostol's text on real analysis, and he has the following theorem: What is the significance/motivation/intuition of this theorem? The proof seems circular/trivial by defining f* ...
0
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0answers
28 views

How to calculate partial derivative of many values?

I have a function for example : $$f(x,y) = x^5 + 3xy + \cos(xy)$$ It's easy to calculate the partial derivative of $x$ or $y$. But how to calculate the partial derivative of $x$ AND $y$, $[ f'x,y ]$
0
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2answers
24 views

Suppose that g is the inverse function of a differentiable function f and G(x) =$\frac{-4}{g^2(x)}$ …

Problem : Suppose that g is the inverse function of a differentiable function f and G(x) =$\frac{-4}{g^2(x)}$ If f(5) =3 and $f'(5) =\frac{1}{125}$ then find $G'(3)$ My approach : f(5)=5 ...
1
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2answers
21 views

Differential equations with Euler's method

A differential equation y' + 2y = 2 - e^(-4*t) With starting point y(0) = 1 and increment ...
-2
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2answers
38 views

Why does $-(3e^{-x})(1-x)-(3e^{-x}) = (-3e^{-x})(2-x)$?

I am looking at an old exam. The first part of the task wants you to differentiate $$ f(x) = 3xe^{-x}, $$ which is $$ f'(x) = 3e^{-x}(1-x) $$ but then, it wants you to differentitate $f'(x)$. While ...
0
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4answers
25 views

Differentiation about the square root of x and y

I stumbled across the curious question and this is how it reads: $\sqrt x + \sqrt y = 16$ Find $\frac{dy}{dx}$. I squared both sides but the result is not conducive to proper differentiation. ...
12
votes
1answer
271 views

About the derivative of a function defined on rational numbers

I have found this problem: Let $f : \mathbb{Q} → \mathbb{R}$ with property: $$|f(x) − f(y)| \le (x − y)^2 \tag1$$ for all $x, y \in \mathbb{Q}$. Prove $f$ is constant. My idea is to consider ...
-1
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2answers
25 views

When the first derivative is increasing slower or faster

This problem is the 3rd question from the GRE practice book (ets.org) here What is the explanation of the answer (C)
-1
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2answers
54 views

Without Extreme Value Theorem, how do we find absolute extrema?

I have to find and classify the critical points of the following functions and then state which relative extrema are absolute extrema. $$f(x,y) = x^3 - y^3 - 2xy + 6$$ $$f(x,y) = xy + 2x - ...
0
votes
0answers
27 views

Finding a formula for a rate of change (x/post from physics.stackexchange)

This is a cross post from the physics stackexchange because I feel like this question is mostly maths anyway and this board is a lot more active. I'm a bit stuck on this question (which is homework ...
11
votes
6answers
736 views

When can we not treat differentials as fractions? And when is it perfectly OK?

I am a first year calculus student so I would prefer if answers remained in Layman's terms. It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat ...
1
vote
1answer
43 views

Convergence of $x_n=f\left(\frac{1}{n}\right), n\geq 1$

I have the following problem to solve: Let $f:]0,1[ \rightarrow R$ be a differentiable function over $]0,1[$, with $|f'(x)|\leq 1, \forall x \in ]0,1[$. (a) Show that the sequence ...
0
votes
1answer
35 views

Elementary Functions, Differentiation, Integration [duplicate]

Why is it that differentiation of a function that is a composition of elementary functions (such as $\sin \:2^x$ or $\ln(\mathrm{arcsec}\: x^3)$ or $x^{1/x}$) always produces a composition of ...
2
votes
0answers
41 views

O'Neill's differential geometry: typo in formula for partial derivative?

I am working through Barrett O'Neill's Elementary Differential Geometry and I'm mildly confused. Exercise 3 in section 4.3 ask you to verify that $$\mathbf{y}_{u}=\mathbf{x}_{u}\frac{\partial ...
2
votes
2answers
35 views

prove or disprove by counter example

If $f$ is differentiable at $x=0$ and $\lim_{x\to0}{f(x)\over x}=3$ then $f(0)=0$ and $f'(0)=3 $. so after a few failed counter examples I decided to prove this since it also kinda seems to be ...
0
votes
0answers
55 views

What does this notation mean? $∂df^2 / ∂x$

I am given variance $\sigma_x$ and function $y=f(x)$ According to my book, the following equation gives the new variance of $y=f(x)$. But I'm not sure what this notation means, as in what this ...
29
votes
9answers
2k views

How is the derivative truly, literally the “best linear approximation” near a point?

I've read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving ...
0
votes
0answers
10 views

show that $\nabla^2 G = \delta (\underline{r} - \underline{r}_0)$ where G is the 2D Green's Function,

I know how to show that $\nabla^2 G = 0$ for $ x \neq x_o, y \neq y_0$ but I don't know how to show that it's $\delta (x - x_0) \delta (y - y_0)$ for $ x = x_0, y = y_0$. Note that I'm using $ G = ...
4
votes
2answers
53 views

Functions and Derivatives

Generaly curious: Let there be a set of functions: Will the sum of the derivatives of the functions be equal to the derivative of the sums?
2
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0answers
23 views

Raising index on covariant derivative

So suppose $X$ is some vector field and $t$ is a tangent vector to some curve on some smooth manifold. Then $t^a\nabla_a X$ gives the directional derivative of the vector field in the direction of ...
1
vote
5answers
94 views

Applications of $f(x_0)=f'(x_0)$

If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$. And where $f''(x_0)=0$ there is an inflection point at $x=x_0$. I am asking are there any ...
0
votes
1answer
32 views

Radon-Nikodym derivative as a Martingale

Let $(\Omega,\mathscr{F}, P)$ be a probability space, let $\nu$ be a finite measure on $\mathscr{F}$, and let $\mathscr{F}_{1}$, $\mathscr{F}_{2}$,... be a non-decreasing sequence of $\sigma$-fields ...
1
vote
2answers
70 views

Limits and differentiability using L'Hospital's rule

Let $y=f(x)$ be an infinitely differentiable function on real numbers such that $f(0)$ is not equal to 0 and $d^n(y)∕dx^n$ not equal to zero at $x=0$ for $n=1,2,3,4$. If $$\lim_{x→0} {f(4x) + af(3x) ...
0
votes
1answer
99 views

Help:to prove increasing function (formula and curves plot)

I am stuck on figuring out why the following function is a increasing function when I read a paper. The function is following $$f(x)=\frac{(x-1)\cdot c^x-2x\cdot c^{x-1}\,\,+x+1}{2^x-x-1}$$, where ...
4
votes
3answers
195 views

Proving $\frac{d}{dx}x^2=2x$ by definition

I did the following proof earlier and just wanted conformation as to whether it works. The question was to show $$\frac{d}{dx}x^2=2x$$ by the difference-quotient definition of a derivative, and then ...
1
vote
1answer
168 views

Help : How to prove the following simple function is decreasing function?

I am stuck on figuring out why the following function is a decreasing function when I read a paper. The function is following $f(x) = \frac{c^x-c\cdot x+x-1}{2^x-x-1}$, where $1<c<2$, and the ...
2
votes
2answers
74 views

The notion of “infinitely differentiable”

Wiki takes me to the section "smoothness" which I don't entirely get, it's just too much stuff for me. My question is, what exactly is it? An infinitely differentiable function is one that is ...
3
votes
2answers
58 views

Multiplying A Coefficient by an Indexed Multiplier using Generating Functions

If I have a particular exponential generating function, $$G(x)=\sum_{n=0}^\infty a_n\frac{x^n}{n!}$$ then what would be the generating function for $$H(x)=\sum_{n=0}^\infty ...
3
votes
1answer
26 views

Partial Derivative with product & chain rule

I cannot for the life of me work out the answer to this partial derivative. $$\frac{\delta}{\delta x}\left(\frac{x^2}{(x+y)^2(x+z)^2}\right) $$ My first thought was: Split into two equations: ...
0
votes
1answer
41 views

A question on derivatives

Let $\lambda \in \mathbb{C}$ and $w_j \in \mathbb{R}$ for all $j$. Let $w(\lambda ) = \sum\limits_{j = 0}^m {{w_j}{\lambda ^j}} $. Is it true that, $w'(\left| \lambda \right|) = \sum\limits_{j = ...
-3
votes
1answer
36 views

$f(x)=-(x-2k)^2(x-4k)-k\quad$ where $k\gt 0\quad$ [closed]

$f(x)=-(x-2k)^2(x-4k)-k\quad$ where $k\gt 0\quad$ (i) Show that the local max occurs at $\left(\frac{10k}3,\frac{32k^3}{27}-k\right)\quad$. (ii) If the local max point is to occur on the x-axis, ...
0
votes
2answers
21 views

Why has $p(\lambda)$ exactly one positive zero?

Let $p(\lambda ) = {a_m}{\lambda ^m} + {a_{m - 1}}{\lambda ^{m - 1}} + \cdots + {a_1}\lambda - {a_0}$ and ${a_0} > 0,{a_1},{a_2}, \ldots ,{a_m} \ge 0$, and at least one of the coefficients ...
2
votes
1answer
29 views

Maximizing sum of logarithms (Z-channel capacity)

In the context of information theory, I am trying to maximize the following function (mutual information of the Z-channel's input and output) with respect to $p$ in order to derive Z-channel's ...
0
votes
1answer
27 views

First and second derivative of summation with inverse tangent

I have the following following function and I would like to learn how to calculate the first and second derivatives with respect to $a$. $$\left[\frac{\sum_{i=1}^n ...
2
votes
1answer
99 views

Application of calculus in real life

I'm no mathematician, so bear with simplicity of what I'm asking. My calculus course(post-Soviet country, a while ago) was utter trash. I've recently decided to approach the topic for self eduction. ...
1
vote
1answer
24 views

Cut corners from rectangle to get box with max volume

I've got a rectangle (no informations about the box, volume box etc.). I need to find how much should I cut from the rectangle to get a box with maximum volume, so I need to find $x = ?$ At the ...
0
votes
0answers
23 views

Find the particular solution given the auxiliary conditions

Let $z=f(x+cy)+g(x-cy)$ where $c$ is a constant and the functions $f$ and $g$ are twice differentiable. Deduce the solution of the equation subject to the auxiliary conditions $$z(x,0)=x \text{ and ...
0
votes
3answers
63 views

Use / Don't use Rolle's Theorem

I've got an interesting exercise and I tried to use Rolle's Theorem to prove it. Do you think my prove is good or what should I use to prove it? Exercise : For continuous function $f$ we've got : ...
2
votes
3answers
32 views

How to find the slope of curves at origin if the derivative becomes indeterminate

What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg-- What is the slope of the curve $x^3 + y^3= 3axy$ at origin and how ...
0
votes
1answer
18 views

Path increment, gradient and total differential

I have trouble proving that in any coordinate system the total differential is equal to the inner product of gradient and path increment in that coordinate syste,. $$dU=\vec{\nabla U}.\vec{dr}$$ ...
0
votes
2answers
42 views

Finding tangent line to curve $x^y-y^x+1=0$

As the title said, I need to find tangent to the curve $x^y-y^x+1=0$, in the point $T(1,2)$ but I don't know how to fond derivative of this implicit function. Of course, I think I could use $\ln$ but ...
0
votes
3answers
20 views

Differentiation of subtraction

I've got an exercise to do and I don't really know what to do. Exercise : We've got function $f$, where $f(a) = 0$ and $f'(a)$ exists. Also we got function $g$ which is continuous. Does exist ...
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1answer
24 views

All Derivatives Bounded from Below

Is it possible to construct a function $f:\mathbb{R} \to \mathbb{R}$ such that there is $c>0$ with the property that for each $n$ and each $x \in \mathbb{R}$ we have $f^{(n)}(x) \geq c$? If not, ...
0
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0answers
31 views

How can show the following function is log-concave?

Suppose that $g(x)$ is an increasing function and $0\leq g(x)\leq1$. I was working on a problem and it reduced to show that if $1-g(x)$ is log-concave then $$f(x)=(1-g^a(x))^b, a\geq 1, b,x>0$$ is ...
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1answer
22 views

Derivative of spherical coordinates [closed]

Why are the r dot terms (eg -r^dotsincos in the z derivative) in the derivatives of the spherical coordinates? Differentiation, as I've understood it, is differentiating a function with respect to a ...