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

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4
votes
1answer
251 views

Gradient of sum of products of matrix traces

For a matrix $X \in \Re_{n\times d}$ find the gradient of $\sum_{i,j}[\langle X_{i.},X_{j.} \rangle\operatorname{tr}(X^TA_{ij}X)]$ w.r.t $X$, where $A_{ij}=(e_i-e_j)(e_i-e_j)^T$ using the basis ...
1
vote
1answer
1k views

Differentiation of a natural log with fraction of trigonometric functions

I am starting with differenciation and I stumbled upon the following exercise: Find $\frac{df(x)}{dx}$, where $$f(x)=\ln \left({\cos x + \sin x \over \cos x - \sin x}\right).$$ So I applied the chain ...
15
votes
1answer
293 views

Is there any meaning to an “infinite derivative”?

I've been thinking about this: say you have an infinitely differentiable function. Then you can form a sequence $f(x), f'(x), f''(x), \cdots, f^{(n)}(x), \cdots$ and attempt to take its limit. For ...
0
votes
2answers
70 views

related rate problem

two fast ants start traveling from the same point in the sand. one ant heads north at a rate of 2cm/s and the other Ant travels east @ a rate of 3cm/s. the trails the ant leave form the sides of a ...
2
votes
1answer
89 views

Derivative of $(Y-HX)^\top C(Y-HX)$ by $X$

I'm trying to derive an expression for $$\nabla_X(Y-HX)^\top C(Y-HX).$$ $Y$ and $X$ are column vectors of size $N \! \times 1$. $H$ and $C$ are matrices of size $N \! \times N$. I have checked this ...
7
votes
1answer
121 views

Can we inverse the mean values theorem?

The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Can we inverse this situation, i.e., given the function $f$ and the real $c$, can we find ...
1
vote
1answer
49 views

Looking for help understanding the asymptotic expansion of the digamma function

I was recently given an example using this asymptotic expansion of the digamma function where: $$\frac{d}{dx}(\ln\Gamma(x)) = \psi(x) \sim \ln(x) - \frac{1}{2x} - \frac{1}{12x^2}$$ Here's the ...
0
votes
2answers
26 views

2nd and 3rd derivative of $A^{-1}$

Consider $$ f\colon R^{n\times n} \to R^{n\times n}, A \mapsto A^{-1}. $$ How can I find the 2nd and 3rd derivative of $f$ applied to $H$, $f'(A)[H]$ and $f''(A)[H,H]$ expanding $f(A+H)$ and ...
1
vote
1answer
38 views

What are definite integrals that use functions in one or both of its limits?

I've seen these types of questions before, but I think I missed a formal explanation of them. I have a solution to the question in front of me, so my question is not related to what the answer is - ...
10
votes
2answers
323 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
1
vote
2answers
1k views

Find the coordinates of any stationary points on the curve $y = {1 \over {1 + {x^2}}}$ and state it's nature

I know I could use the quotient rule and determine the second differential and check if it's a max/min point, the problem is the book hasn't covered the quotient rule yet and this section of the book ...
1
vote
3answers
553 views

Find dy/dx given $y = {1 \over {1 + \sqrt x }}$, using the chain rule

This is what I tried: $$ {1 \over {1 + \sqrt x }} = {1 \over {\sqrt {1 + x} }} = {1 \over {{{(1 + x)}^{{1 \over 2}}}}} = {(1 + x)^{ - {1 \over 2}}} $$ $${{dy} \over {dx}} = - {1 \over 2}{(1 + x)^{ - ...
2
votes
2answers
118 views

If derivative $f'$ of a function $f$ satisfies $0 < C \leq f'(x)$ for all $x$ then $f$ is bijective

Proposition If derivative $f'$ of a function $f:\mathbb R\to\mathbb R$ satisfies $0 < C \leq f'(x)$ for all $x\in\mathbb R$ then $f$ is bijective. It is clear that if there exist ...
2
votes
3answers
113 views

Linear approximation to 1/0.254

The question says: Use linear approximation to approximate $1/0.254$. I know that $1/0.25 = 4$. Where do I proceed from next. Do I subtract $0.004$ from the answer, or what else could I do?
3
votes
2answers
54 views

Graph Concavity Test

I'm studying for my final, and I'm having a problem with one of the questions. Everything before hand has been going fine and is correct, but I'm not understanding this part of the concavity test. ...
4
votes
1answer
116 views

Need help understanding if a function is increasing or decreasing

I am working on understanding the following function: $$g(x) = \ln\Gamma\left(\frac{x}{4}\right) - \ln\Gamma\left(\frac{x}{5}+\frac{1}{2}\right) - \ln\Gamma\left(\frac{x}{20}+\frac{1}{2}\right) - ...
2
votes
1answer
358 views

Differentiation under the Integral sign for the Lebesgue integral

I want to prove the following version of Liebniz's Rule: Let $f:[a,b]\times [c,d]\to \mathbb{R}$ be integrable with respect to the first variable, $\phi,\psi:[c,d]\to [a,b]$ be differentiable and let ...
1
vote
0answers
279 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
2
votes
1answer
88 views

The rate of increase of the Gamma Function over real numbers

If $$ x_1 > x_2 > 0$$ and $$\Delta{x}>0$$ does it follow that: $$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$ Would it be enough to show ...
0
votes
1answer
19 views

Vector derivative partial to w

What's the derivative of E (below equation) partial to w: ( consider that w is vector)
4
votes
1answer
143 views

Find the order of the error for the approximation $f' '(x)$

Given $$f''(x) = \frac{ f(x+h) - 2f(x) + f(x-h)}{h^2}.$$ I realize that this is just an approximation - that it won't give the exact value of $f''(x)$ and therefore there is an error term. However, I ...
0
votes
3answers
120 views

If $a,b$ are extended real numbers, $f$ is differentiable/$f'$ is continuous on $(a,b)$, prove uniform continuity.

Suppose $a<b$ are extended real numbers and that $f$ is differentiable on $(a,b)$. Prove that if $f'$ is bounded on $(a,b)$ then $f$ is uniformly continuous on $(a,b).$
2
votes
1answer
51 views

Differentiability of first derivative of a function

If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples? From Darboux ...
8
votes
5answers
4k views

Using the Limit definition to find the derivative of $e^x$

I was wondering how we could use the limit definition $$ \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ to find the derivative of $e^x$, I get to a point where I do not know how to simplify the ...
1
vote
0answers
111 views

Chain rule for function of two derivatives

Let $u = f(x, y)$ be a $C^2$-function. Let $x = rcos\theta$ and $y = rsin\theta$. Compute $\frac {d^2u}{dr^2}$ in terms of partial derivatives. Answer: $\frac {d^2u}{dr^2}$ = $\frac ...
4
votes
3answers
345 views

Using L'Hôpital's rule to find $\lim_{x\to\pi/2}(\tan x)(\ln \sin x)$

I have $$\lim_{x\to\pi/2}(\tan x)(\ln \sin x)$$ And I need to solve it using L'Hôpital's rule. I can turn it around to get $\;\; (0/1)\cdot0$ But I don't see how to get $0/0$ to move on to the ...
2
votes
2answers
707 views

Lagrange polynomial and derivative problem

I need to code this function in matlab $$L'(x) = \sum_{k = 0}^{n} f_k l_k(x) $$ Where's $l_k$ looks like this $$l_k(x) = \sum^{n}_{j=0, j \neq k} \frac{\prod^{n}_{i=0}(x - x_i)}{(x - ...
1
vote
2answers
52 views

Taylor series $ \sqrt{\frac{t}{t+1}}$

Could someone tell me how to calculate $ \sqrt{\frac{t}{t+1}}$ it should be $ \sqrt t - \frac{t^{\frac{3}{2}}}{2} +O(t^{\frac{5}{2}}) $
2
votes
3answers
53 views

Solving $(f'(x))^2 = f(x)f''(x)$ with boundary conditions.

Let $f$ be a continuous real-valued function such that $$(f'(x))^2 = f(x)f''(x).$$ Suppose $f(0) = 1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$. I have this question in my book ...
3
votes
3answers
160 views

Calculation of a derivative

I have to calculate the following derivative $$\frac{\partial}{\partial{\Vert x\Vert}}e^{ix\cdot y}$$ Then I write $$e^{ix\cdot y}=e^{i\Vert x\Vert\Vert y\Vert\cos\alpha}$$ andI derive; is this ...
4
votes
4answers
2k views

Show that function is strictly monotone increasing

I want to show that $$ f(x)=\dfrac{x-\sin(x)}{1-\cos(x)} $$ is strictly increasing in $(0,2 \pi) $. Unforunately, this is not that easy for me , as the derivative is not very manageable and ...
2
votes
2answers
1k views

Finding the 9th derivative of $\frac{\cos(5 x^2)-1}{x^3}$

How do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series, ...
0
votes
1answer
50 views

Finding domain of the sum of a series function

a. Find the domain of the definition $D \subset \mathbb R$ for the function $$f(x)=\sum_{n=1}^\infty (-1)^n{x \over n+x}$$ b. For what values of $x\in D$ the function $f$ is differentiable? My ...
1
vote
1answer
64 views

Derivative of $ C_m = \frac{{|x|}e^{2x}}{\pi^{1/2}}\int_0^t \alpha ^{-\beta/2}e^{\frac{-x^2}{\alpha}-\alpha } \mathrm d\alpha $

I have a expression like this $$ C_m = \frac{{|x|}e^{2x}}{\pi^{1/2}}\int_0^t \alpha ^{-\beta/2}e^{\frac{-x^2}{\alpha}-\alpha }d\alpha $$ I need to compute the values of $\partial C_m /\partial x$. ...
1
vote
1answer
47 views

Why no roundoff error when dividing by h in this approximation

Why the approximation $f'(x) = ( f(x+h) - f(x-h) ) / 2h$ produces no roundoff error if $h = 2^{-k}$, where $k$ is any integer. Thanks very much in advance.
3
votes
4answers
140 views

$f'$ odd implies that $f$ is even

I can prove that if $f$ is even then $f'$ is odd (and $f$ odd implying $f'$ is even). However, I'm not sure how to prove it the other way around by using limits. To prove that $f'$ being odd implies ...
2
votes
2answers
147 views

If $a_1,a_2,\dotsc,a_n>0 $, then $\lim\limits_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+\dotsb+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n$

If $a_1,a_2,\dotsc,a_n $ are positive real numbers, then prove that $$\lim_{x \to \infty} \Bigl[\frac {a_1^{1/x}+a_2^{1/x}+.....+a_n^{1/x}}{n}\Bigr]^{nx}=a_1 a_2 \dotsb a_n.$$ My Attempt: ...
0
votes
2answers
113 views

Let $f:\Bbb R^2→\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$

Let $$f:\Bbb R^2\to\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$$ i) Is $f$ continuous at $(0,0)$? ii) Is $f$ differentiable at $(0,0)$? I can prove that $f$ is ...
1
vote
1answer
61 views

Find $f^{(n)}(x)$ for $f(x) = 5x^4-8x^3+6x^2-1$

I'm a bit lost and how I would go about creating a general formula for differentiating this equation. Find $f^{(n)}(x)$ for $$f(x) = 5x^4-8x^3+6x^2-1.$$
3
votes
1answer
493 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 ...
1
vote
3answers
193 views

Implicit Differentiation Problem???

PROBLEM: Heat flows normal to isotherms, curves along which the temperature is constant. Find the line along which heat flows through the point $(2,5)$ when the isotherm is along the graph of ...
3
votes
2answers
166 views

Matrix Derivative of $ABC$ with respect to $B$

I have looked throughout the matrix cookbook and other sources, but am a bit confused by this problem. If I have a function $F = ABC$, what is the partial derivative of $F$ with respect to $B$? When ...
0
votes
2answers
116 views

Is my solution correct for the derivative of $\frac{\sin 3x}{\sqrt{3}}$?

Is my solution for the derivative of $\frac{\sin 3x}{\sqrt{3}}$ correct? $$\begin{align} &\frac{d}{dx} \frac{\sin 3x}{\sqrt{3}} \\ & = \frac{0 \cdot \sin3x - \sqrt{3} \cdot 3\cos ...
0
votes
1answer
80 views

Differential equation solutions and change of variable

Consider the differential equation $$x''+\frac{a}{t}x'+\frac{b}{t^2}x=0$$ for $t>0$ where $a,b\in\mathbb{R}$ are constants. I need to show that $x:\mathbb{R^+}\to\mathbb{R}$ is a solution of this ...
2
votes
3answers
142 views

Derivative of $\sin^2 (\sqrt{t})$

I need to find the derivative of $\sin^2 (\sqrt{t})$ which I believe have done but the answer seems to be more simplified and I don't know how to arrive to it. Here are my steps $$\begin{align} & ...
2
votes
2answers
98 views

Why is $\frac{f'(x)}{f(x)}$ always a constant?

Today in class we learned that for exponential functions $f(x) = b^x$ and their derivatives $f'(x)$, the ratio is always constant for any $x$. For example for $f(x) = 2^x$ and its derivative $f'(x) = ...
2
votes
2answers
238 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}}) - ...
0
votes
0answers
123 views

Math Analysis - Problem with finding domain of series of functions

a. Find the domain of the definition $D \subset \mathbb R$ for the function $$f(x)=\sum_{n=1}^\infty (-1)^n{x \over n+x}$$ b. For what values of $x\in D$ the function $f$ is differentiable? Well I ...
2
votes
4answers
733 views

Using integration and differentiation to solve for deceleration?

PROBLEM STATES: The landing velocity of an airplane (at which it touches the ground) is 100mi/hr. It decelerates at a constant rate and comes to a stop after traveling .25miles along a straight ...
2
votes
0answers
135 views

Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...