Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
0
votes
0answers
12 views
$L:\mathbb{R}^2\to\mathbb{R}^2$ given by $L(x,y)=(x,-y)$ which of the following is true?
$L:\mathbb{R}^2\to\mathbb{R}^2$ given by $L(x,y)=(x,-y)$ which of the following is true?
differentiable everywhere on $\mathbb{R}^2$
differentiable on $(0,0)$ only
$DL(0,0)=L$
$ DL(x,y)=L$ for all ...
0
votes
0answers
16 views
calculate derivative for standard inner product
$L\colon\mathbb{R}^n\to \mathbb{R}$ $L_y(x)=\langle x,y\rangle$ for some inner product, $DL$ be the derivative of $L$. Its a Linear map so I know derivative will be itself only.
I want to calculate ...
0
votes
0answers
38 views
Question on derivative
I want to differentiate $H(p(t),q(t))=1 $ with respect to $t$, where $H:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} $ is a convex function.
I think that it is:
$\displaystyle \frac{dp}{dt} ...
1
vote
0answers
19 views
Derivative methods for artifical neural networks with single hidden layer
I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
0
votes
0answers
9 views
what is the derivative of Berthelot combining rule
Is anyone knows the derivative of Berthelot combining rule
$\epsilon_{ij}$ = $\sqrt {\epsilon_{ii} \epsilon_{jj}}$
Thanks in advance
1
vote
1answer
28 views
Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)
Got a quick question from a past exam paper.
If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
2
votes
1answer
50 views
derivative of square root of $x_1$ and $x_2$
I am confusing with calculation of derivative of $$\sqrt{x_1x_2}$$ I am very thankful if anyone help me out of this problem.
my question is that,
(1) $x_1$ and $x_2$ are two different variable i just ...
3
votes
2answers
31 views
sum of polynoms of given property
I have $P(x)$ a polynomial with degree $n$ ,$P(x) \ge 0$ for all $x \in$ real.
I have to prove that:
$f(x)=P(x)+P'(x)+P"(x)+......+P^{n}(x) \ge 0$ for all $x$.
I tried different methods to ...
6
votes
3answers
107 views
First derivative of $\sqrt[\large 5]{\frac{t^3 + 1}{t + 1}}$
I have yet another derivative I need help with. I have to differentiate :
$$\sqrt[\uproot{3}{\Large 5}]{\frac{t^3 + 1}{t + 1}}$$
with respect to $t$.
I had two thoughts about this, use the chain ...
0
votes
1answer
44 views
Geometric representation of product rule?
At time 1:06 of this video by minutephysics, there is a geometric representation of the product rule:
However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot ...
1
vote
1answer
18 views
Differentiability of functions of several variables.
I've just read the proof of a theorem which states that if a function of several variables(two in this case) has partial derivatives in some neighborhood of a point (x,y) and these derivatives are ...
0
votes
1answer
30 views
Binomial sum of derivatives
I would like to know the result of the following sum:
$$\sum_{p=0}^m \binom{m}{p}(-1)^{p-1}\frac{\partial^{p-1}}{\partial x^{p-1}}f(x)\cdot(-1)^{m-p-1}\frac{\partial^{m-p-1}}{\partial ...
0
votes
0answers
12 views
A general formula for the partial derivatives of $\sigma(\xi_1,\ldots,\xi_j,-(\xi_1+\cdots+\xi_m),\xi_{j+1},\ldots,\xi_{m})$.
Let $\sigma$ be defined on $(\mathbf{R^n})^m\backslash \{0\}$ and suppose it is adequately differentiable (that is, we can take as many derivatives as required to show this next statement). If ...
10
votes
4answers
109 views
Closed form for $n$th derivative of exponential of $f$
What is the closed form for:
$$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
0
votes
1answer
96 views
Changing order of derivatives
I would like to rewrite the following expression
$$\frac{d^i}{dx^i}\left\{f(x)\left[\frac{d^jf(x)}{dx^j}\right]\left[\frac{d^kf(x)}{dx^k}\right]\right\}$$
into the form
$$D f(x)^3,$$
with $D$ ...
10
votes
2answers
188 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
2
votes
3answers
41 views
derive using the chain rule
Given the polinomyal $f(x)=\frac{x^3}{(4-x^2)^3}$ find $f'(x)$
So, If I try to derive this, first I must to apply the chain rule in the denominator and then derive of the division (...)
...
0
votes
1answer
21 views
Application of derivative - how to calculate change in error
Problem :
If the error committed in measuring the radius of the circle is $0.05\%$ then find the corresponding error in calculating the area.
Solution : Let the error can be denoted by $\delta r = ...
1
vote
1answer
22 views
Differentiation problem of power to infinity by using log property
Problem:
Find $\frac{dy}{dx}$ if $y =\left(\sqrt{x}\right)^{x^{x^{x^{\dots}}}}$
Let ${x^{x^{x^{\dots}}}} =t. (i)$ Taking $\log$ on both sides $ \implies {x^{x^{x^{\dots}}}}\log x = \log t$
This ...
0
votes
0answers
22 views
Equation of tangent at point P(t) where t is any parameter
Problem :
Find the equation of the tangent at the point P(t), where t is any parameter, to the parabola $y^2 = 4ax$
I have the solution it states the coordinate of the points are ($at^2,2at)$ ...
3
votes
1answer
38 views
Closed form for $n$-th derivative of exponential
I need the closed-form for the $n$-th derivative ($n\geq0 $):
$$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$
Thanks!
By following the suggestion of Hermite polynomials:
...
1
vote
1answer
25 views
Application of derivative - Want to know the relation of graph with this problem
Problem :
For the curve $y = 3 \sin \theta \cos \theta, x = e^\theta \sin \theta, 0 \leq \theta \leq \pi $ ; the tangent is parallel to x-axis when $\theta$ is
a) $0$
b) $\dfrac{\pi}{2}$
c) ...
1
vote
1answer
52 views
Is the following differentiating under the integral sign correct?
Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial ...
4
votes
1answer
31 views
Complexified tangent vector, complex manifold
Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
2
votes
2answers
43 views
Find the minimum of $|a+\frac 2 {a-1}|$ where $|a|\leq2$.
Find the minimum of $|a+\frac 2 {a-1}|$ where $|a|\leq2$.
I tried using differentiation, but the absolute makes things troublesome...
Please help. Thank you.
2
votes
1answer
78 views
Derivative of (f(x+dx)-f(x))
Am I correct to state that taking the derivative of $f(x+dx) - f(x)$ with respect to $x$ does not equal $df/dx$? Or in another words:
$$\frac{d}{dx}(f(x+dx) - f(x)) \neq \frac{df(x)}{dx}$$
I've ...
-1
votes
0answers
55 views
Please take the time to double check my working for a calculus question! Please!
The Link below is the question sheet and these are my answers. I'm very unsure as to whether the last two are correct or not... Help please!
http://tinypic.com/r/69n7sg/5
Are my answers correct?
...
1
vote
1answer
29 views
Small question about derivative
how to derive $\int_0^1 G(t,s) e(s)ds$ with respect to $t$
Where $G(t,s)$ is a Green function and $e:(0,1)\rightarrow \mathbb{R}$ continuous and $e\in L(0,1)$
Please help me
Thank you
1
vote
2answers
66 views
A simple proof about $e^x$?
Do you guys think this is correct? I am trying to prove that there is no single-term polynomial function (oxymoron, I know) $f(x)$ which is always (or at least as x approaches infinity) greater than ...
2
votes
2answers
48 views
Suppose $f$ is a real-differentiable function on $[a,b]$ and suppose $f'(a)<c<f'(b)$. Prove then there is a point $x \in (a,b)$ such that $f'(x)=c$
This is what i have:
Put $g(t) = f(t) - ct$.
Then $g'(a)<0$ so that $g(t_{1}) < g(a)$ for some $t_{1} \in (a,b)$ and
$g'(b)>0$ so that $g(t_{2}) < g(b)$ for some $t_{2} \in (a,b)$.
...
2
votes
1answer
16 views
Let $f$ be defined on $[a,b]$, Prove that if f has a local maximum at a point $x \in (a,b)$, and if $f'(x)$ exists, then $f'(x)=0$
Is this proof correct:
Let's choose a $\delta$ to that
$a < x - \delta < x < x + \delta < b$
If $ x - \delta < t < x$
then $\frac {f(t) - f(x)} {t-x} \geq 0$
Letting $t ...
2
votes
1answer
29 views
Find expression for $dy/dx $ + state where it is valid
hopefully you guys can shed some insight into this question I'm working on.
Given
$xy+y^{2}-e^{x^{2}} = 6$
find an expression for $dy/dx$ and state where it is valid.
So, what I did was ...
3
votes
2answers
65 views
Why is this derivative incorrect?
We have to find the derivative of $$f(x) = \dfrac{\tan(2x)}{\sin(x)}$$
I would like to know why my approach is incorrect:
$$f'(x) = \dfrac{\sin(x) \cdot \dfrac{2}{\cos^2(2x)} - \tan(2x) \cdot ...
0
votes
1answer
25 views
Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically
$\eqalign{
& y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr
& \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr
& {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 ...
2
votes
3answers
40 views
Second Derivative of basic fraction using quotient rule
I know this is a very basic question but I need some help.
I have to find the second derivative of:
$$\frac{1}{3x^2 + 4}$$
I start by using the Quotient Rule and get the first derivative to be:
...
1
vote
3answers
59 views
How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?
Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.
I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
0
votes
2answers
46 views
Graphing $y= \frac{x^3}{x^2-1}$
I'm having a lot of problems trying to graph this... other than using a graphics calculator!
I know the domain is all real values of $x , x$ does not equal $-1 \text{or} 1$. The point of inflection ...
4
votes
4answers
125 views
Differentiate $\log_{10}x$
My attempt:
$\eqalign{
& \log_{10}x = {{\ln x} \over {\ln 10}} \cr
& u = \ln x \cr
& v = \ln 10 \cr
& {{du} \over {dx}} = {1 \over x} \cr
& {{dv} \over {dx}} ...
1
vote
1answer
21 views
Finding the value of t where tangent line is perpendicular to x axis
For the curve x = t$^2 - 1, y = t^2 - t$, the tangent line is perpendicular to x-axis, where
Options are :
a ) t = 0
b) $t \to \infty$
c) $t = \frac{1}{\sqrt{3}}$
d) $t = \frac{-1}{\sqrt{3}}$
...
5
votes
2answers
57 views
Inverse and derivative of a function [duplicate]
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
4
votes
2answers
129 views
Is $y=|x^3|$ a smooth function?
Is this a smooth function? $y=|x^3|$
The graph of this function has no sharp cuts or corners, so I think it is a smooth function but someone told me that it's not.
7
votes
1answer
94 views
What is this limit called? Is it a different kind of derivative?
(first I should notice you this is not something I can look up in a textbook, because I'm learning partial derivatives, alike I do with most Maths, as a hobby. If something below is wrong, blame the ...
1
vote
2answers
30 views
Relationship between second order derivatives and cross derivative of smooth surfaces
Probably a silly question, but I wonder if $z=f(x,y)$ is a smooth surface, and the values of its two second order derivatives $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ ...
1
vote
0answers
37 views
Math question derivatives related?
So I have to find the constant c in order that the function $\displaystyle z= x^c e^{-y/x}$ proves the equation in the image
The problem is that I don't understand what $\displaystyle z_{yy}$ mean ...
0
votes
2answers
26 views
Math question partial derivatives
I have to find the $\partial^2 z/\partial x \partial y$ of $z=e^{xy}$. I know how to find $\partial^2 z/\partial x^2$ which by the way is $y^2 e^{xy}$ but not this one...can you give me a little ...
1
vote
2answers
36 views
Derivative of Trig Functions (Intuition Help?)
Looking for some intuition help here.
I have the following exercise and these are the steps I take:
$$
y = \sin\left(\frac{1}{x}\right)
$$
$$
u=\frac{1}{x}
$$
$$
y = \sin u,\;\;\frac{dy}{du} = \cos ...
2
votes
1answer
28 views
Derivative with multiplication and division
So I have the following homework. I don't want the answer, only point me in the right direction please. Thanks.
I'm stuck in the product rule. Do I apply the product rule twice or just one time after ...
1
vote
3answers
62 views
existence of a derivative $x\cdot f(x)$
Lets say we have a function $f(x)$ that has a derivative at point $a$.
Can we prove that the function $x\cdot f(x)$ has also a derivative at point $a$?
If this is not true, can anybody give an example ...
0
votes
0answers
31 views
Proof the theorem $\int_0^\infty {{t^n}f(t)dt = {{( - 1)}^{n + 1}}\int_0^\infty {{F^{(n + 1)}}(u)du} } $
Knowing that:
$$ L\left[ {\int_0^\infty {\frac{{f(t)}}{t}dt} } \right] = \frac{1}{s}\int_0^\infty {F(u)du}$$
with: $L[f(t)] = F(s) $
Show that:
$$\int_0^\infty {\frac{{f(t)}}{t}dt ...
3
votes
2answers
63 views
Finding the Derivative without using Product or Quotient Rule
I have a math problem where I am required to find the derivative of a function with the limitations of not being allowed to use the Product or Quotient Rule of Differentiation.
The problem looks like ...
