Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
39
votes
1answer
1k views
How discontinuous can a derivative be?
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
20
votes
6answers
798 views
Uses of $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$
I have been wondering whether the following limit is being used somehow, as a variation of the derivative:
$$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$
Edit:
I know that this limit is defined in ...
3
votes
1answer
260 views
Partial derivative confusion.
I don't understand partial derivatives. Here's an example that nails down my confusion:
Suppose we have some variables $x$, $p$, and $q$ with $p=x^2$ and $q=e^x$. Then $$\frac{\partial q}{\partial p} ...
3
votes
2answers
1k views
What is the formula for nth derivative of arcsin x, arctan x, sec x and tan x?
Are there formulae for the nth derivatives of the following functions?
1) arcsin x
2) arctan x
3) sec x, and
4) tan x
Thanks.
7
votes
1answer
196 views
Inverse of a bijection f is equal to its derivative
Does there exist a differentiable bijection $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) = f^{-1}(x)$ ?
6
votes
1answer
566 views
“Strong” derivative of a monotone function
It is well known that if a function $f\colon \mathbb{R} \to \mathbb{R}$ is monotone then $f'$ exists almost everywhere.
Is it true that if $f$ is monotone then there exists (edit: I mean exists ...
3
votes
3answers
1k views
Second derivative “formula derivation”
I've been trying to understand how the second order derivative "formula" works:
$$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$
So, the rate of change of the rate of change for an arbitrary ...
26
votes
5answers
864 views
Are there ways of finding the $n$-th derivative of a function without computing the $(n-1)$-th derivative?
Say we have a function $f(x)$ that is infinitely differentiable at some point.
Is it possible to find $f^{(n)}(x)$ without having to find first $f^{(n-1)}(x)$? If so, does it take less effort than ...
15
votes
1answer
344 views
“Converse” of Taylor's theorem
Let $f:(a,b)\to\mathbb R$. We know that for every $c\in(a,b)$ we can write $f(t)=\sum_{i=0}^k a_i(c)(t-c)^i+o\left((t-c)^k\right)$ and $\forall i$ $a_i(c)$ is continuous (with respect to $c$). Can we ...
4
votes
4answers
2k views
Derivative of Integral
I'm having a little trouble with the following problem:
Calculate $F'(x)$:
$F(x)=\int_{1}^{x^{2}}(t-\sin^{2}t) dt$
It says we have to use substitution but I don't see why the answer can't just be:
...
4
votes
1answer
2k views
Understanding the relationship between differentiation and integration
I am trying to understand the relationship between differentiation and integration. Differentiation has been introduced to me by this diagram:
Which displays that the derivative of a point $x$ on ...
8
votes
5answers
639 views
Calculating the shortest possible distance between points
Question:
Given the points $A(3,3)$, $B(0,1)$ and $C(x,0)$ where $0 < x < 3$, $AC$ is the distance between $A$ and $C$ and $BC$ is the distance between $B$ and $C$. What is x for the distance ...
6
votes
5answers
660 views
Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$
I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of ...
3
votes
2answers
370 views
How to disprove this fallacy that derivatives of $x^2$ and $x+x+x+\cdots\ (x\text{ times})$ are not same. [duplicate]
Possible Duplicate:
Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition)
$$x^2=\underbrace{{x+x+x+\cdots+x}}_{x \text{ times}}$$
$$\therefore \frac ...
3
votes
6answers
559 views
Question about finding the limit at an undefined point.
This may be braindead, but I'm trying!
If I have a function $f$ and that function is not defined at some x, then asking for the derivative of the function at $x$ makes no sense since there is no ...
2
votes
2answers
104 views
Derivative of an integral
I would like some guidance on how to solve these type of problems.
Find $h'(x)$ if $$h(x) = \int\limits_{\cos(x)}^x \mathrm{e}^{t^2} \, dt$$
Mathematica says $h'(x) = e^{x^2} - e^{\cos^2(x)} ( - ...
1
vote
4answers
217 views
Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I'm looking for a way to find this limit:
$\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I think I have found that it diverges, by plugging numbers into the formula and "sandwich" the result. However I ...
0
votes
2answers
118 views
Can anyone help me to differentiate this equation?
I need to differentiate this equation with respect to $x$:
$$
...
0
votes
4answers
290 views
Show the negative derivative of a function.
A type of interaction between atoms in a molecule is called a Van der Waals interaction. This can be described by the potential energy function;
$$U= ...
0
votes
1answer
167 views
Evaluate the partial derivatives of the following function:
The function $f: \mathbb{R} \longrightarrow \mathbb{R}$ is defined by the rule
$$f(x,y) = \begin{cases} \frac{x^5y}{x^4+y^2}, & (x,y) \neq (0,0), \\ 0, & (x,y)=(0,0). \end{cases}$$
...
29
votes
1answer
474 views
How to show that $f'(x)<2f(x)$
I would appreciate if somebody could help me with the following problem:
Q: Let $f(x),f'(x),f''(x),f'''(x)>0$ , $f'''(x)$ is a continuous function and $f'''(x)<f(x)$ on $\mathbb{R}$
then ...
7
votes
3answers
385 views
Finding the $n$-th derivatives of $x^n \ln x$ and $\frac{\ln x}{x}$.
How can I prove the following identities:
$$ \left( x^n \ln x \right)^{(n)}= n! \left(\ln x + 1 + \frac{1}{2} +\cdots +\frac{1}{n} \right), \quad x>0, \quad n\ge 1, \tag{a}$$
$$ \left( \frac{\ln ...
3
votes
1answer
400 views
Matrix calculus : Find the gradient/derivative?
I know that the derivative of $Tr(Z^TAZ)$ w.r.t $Z$ is $2AZ$. Now I'd like to compute the derivative of $Tr\left[Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)Z\right]$ instead, w.r.t $Z ...
2
votes
0answers
266 views
Prove that sum is finite with the help of generating function
Please help me to prove that the following sum is finite
$$
\sum_{j=2l-2}^{\infty}j!\, a_j^{(l)},
$$
here the generating function of $\displaystyle{a_j^{(l)}}$ is ...
8
votes
3answers
221 views
n-th derivative of $\frac{\ln x}{x}$.
Let $f(x)=\frac{\ln x}{x},x>0$. Show that $$f^{(n)}(1)=(-1)^{n+1}(n!)(1+\frac{1}{2}+\cdot+\frac{1}{n})$$
Trial: n-th derivative of $\ln x$ is $$(-1)^{n-1}(n-1)! x^{-n}$$ and n-th derivative of ...
2
votes
1answer
238 views
Partial derivatives and orthogonality with polar-coordinates
We are stuck with this question here because I cannot understand the following results. I find it hard to visualize this, let alone deduce from that. How to do it?
Objective to Attack The closely ...
0
votes
0answers
89 views
Divergence of the Derivative of the Prime Counting Function
On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written
$$
\pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$
with $ \operatorname{R}(z) = ...
8
votes
2answers
394 views
Sobolev meets Wiener
Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point.
Does it at least have weak derivatives?
6
votes
2answers
1k views
Differentiating an Inner Product
If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
5
votes
2answers
352 views
Why use the derivative and not the symmetric derivative?
The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
5
votes
3answers
261 views
Proof that $Γ'(1) = -γ$?
I know that $Γ'(1) = -γ$, but how does one prove this?
Starting from the basics, we have that:
$$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$
How do we differentiate this? How do we then find that
...
4
votes
4answers
88 views
Can someone give me a deeper understanding of implicit differentiation?
I'm doing calculus and I want to be an engineer so I would like to understand the essence of the logic of implicit differentials rather than just memorizing the algorithm. Yes, I could probably ...
4
votes
2answers
59 views
Prove that $h^{(k)}(0)=\lim_{t\to0}\frac{\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}h(jt)}{t^k}$
Prove that if $h$ is infinitely differentiable in a neighborhood of $0$, then the kth derivative evaluated at 0 is
$$h^{(k)}(0)=\lim_{t\to0}\frac{\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}h(jt)}{t^k}$$
4
votes
2answers
442 views
Proving Thomae's function is nowhere differentiable.
I am given the function
$$f(x)=\begin{cases} 0 \text{ ; when } x \text{ is irrational} \\\frac 1 q \text{ ; for } x=\frac p q \text{ irreducible fraction}\end{cases}$$
Spivak proved that for $a\in ...
3
votes
2answers
201 views
how to calculate $\frac{d\dot{x}}{dx}$
Let $x$ depend on $t$. $\dot{x}$ is derivative x over t. I want to know is there some formulas to simplify $\frac{d\dot{x}}{dx}$? Any hint or thought is appreciated. Thank you!
2
votes
2answers
90 views
Taking a complicated partial derivative
Mostly I believe in math. However I have trouble in my economic textbook (which really should be right).
I have the following equation:
$$ u(c,d)=\left(ac^{\frac{1-\gamma}{\theta}}+b ...
1
vote
1answer
132 views
Commutators, and Christoffel symbols in a non holonomic basis
I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors.
I need to compute ...
1
vote
1answer
102 views
General solution of $ \frac{d^j}{d\sigma^j}(\exp(0.5(\alpha-\sigma)^2) $
How would you write the general solution, I'm assuming something like a sum, of:
$$ \frac{d^j}{d\sigma^j}\exp[0.5(\alpha-\sigma)^2] $$
Regards,
0
votes
1answer
141 views
Show that this piecewise function is differentiable at $0$
I have shown (from first principles) that the Cauchy-Riemann equations for the following function are satisfied at $z=0$. But to properly prove differentiability at $z=0$, what should I do next? Do I ...
0
votes
1answer
56 views
Continuity of a function, Differentiable function
The following function is given:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ x\rightarrow \begin{cases} x^2\cos{\left(\frac{1}{x}\right)} & \text{for } x \neq 0\\ 0& \text{for } x =0\end{cases}$$
...
-3
votes
1answer
240 views
Partial derivative problem
Let
$$f(x,y) = \begin{cases} \dfrac{xy(x^2-y^2)}{x^2+y^2}, & (x,y) \neq (0,0), \\
0, & (x,y)=(0,0). \end{cases}$$
Show that
(A) $f_{xy}(0,0) \neq f_{yx}(0,0)$.
(B) $f$ is differentiable at ...
5
votes
1answer
134 views
problem on continuous and differentiable function
$f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and $f(0) \geq 0$ and $f^\prime(x) \geq f(x)$
to show $f(x)\geq 0 \forall x \in (0,\infty)$
my answer: if $\exists x_0 \in ...
3
votes
3answers
247 views
Discontinuous derivative.
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
3
votes
6answers
157 views
Simple partial differentiation $x = r\cos\theta$ and $y = r\sin\theta$
If
\begin{align}
x &= r\cos\theta,\\
y &= r\sin\theta,
\end{align}
find
$$\dfrac{\partial^2\theta}{\partial{x}\partial{y}}.$$
How can I find this partial derivative?
I need to prove ...
3
votes
4answers
393 views
Are there “one way” integrals?
If we suppose that we can start with any function we like, can we work "backwards" and differentiate the function to create an integral that is hard to solve?
To define the question better, let's say ...
2
votes
2answers
144 views
Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation
I am trying to get the lower bound for:
$f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ...
2
votes
3answers
132 views
Silly question: Why is $\sqrt{(9x^2)} $ not $3x$?
I had to find the derivative of $f(x) = \sqrt{(9x^2)}$. I applied chain rule with the following steps.
Let $f(x)$ be $\sqrt{x}$ and $g(x)$ be $9x^2$
$$ \begin{align} &f'(g(x)) \times g'(x) \\
...
2
votes
1answer
433 views
Partial derivative of integral: Leibniz rule?
The Liebniz rule is as follows:
What I would like to know is how to apply the above formula for the case of the partial derivative:
$\displaystyle \ \ \frac{\partial}{\partial\alpha} ...
2
votes
6answers
201 views
Derivative of $x^2$
This seems too easy, but here's the question:
$x^2$ is $x + x + ...+ x$ (with $x$ terms). Its
derivative is $1 + 1 + ... + 1$ (also $x$ terms). So the derivative
of $x^2$ seems to be $x$.
And ...
2
votes
2answers
252 views
Can the Sum Rule for derivatives be extended to infinite series?
I wrote an answer here, which I'm not sure works.
The sum rule for differentiation of two functions says that $D(u+v) = D(u) + D(v)$ where $D$ indicates the derivative, and $u$ and $v$ two functions. ...
