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

learn more… | top users | synonyms (2)

1
vote
0answers
41 views

$f(x)=\cos(ax+b)$ find $f'(x)$ , $f''(x)$, $f'''(x)$, $f''''(x)$ and give a general form for $f^{(n)}( x)$ where there s cosinus

I found \begin{align*} f'(x)&=-a\cos(\pi/2-ax-b) \\ f''(x)&= a^2 \cos(ax+b) \\ f'''(x)&=-a^3 \cos(\pi/2-ax-b) \\ f''''(x) &= a^4 \cos(ax+b) \\ f^{(n)} (x)&= (-1)^n a^n \cos(\dotsb ...
3
votes
1answer
57 views

Did I draw these derivatives right?

I haven't started finding the derivatives of functions yet, so at the moment this is strictly about finding the right derivative graph to an original graph. The task was this: Look at the graph ...
1
vote
0answers
42 views

Dual quaternion derivation

I'd like to derivate a dual quaternion \begin{align} \hat{q}&=(1 + \frac{1}{2}\epsilon\vec{t})q \end{align} where \begin{align} q &= e^\vec{w} , \\\vec{w}&=(0, w_1,w_2,w_3)^t ...
1
vote
0answers
48 views

Continuity and differentiability of $x^a\sin ({1\over x}) $ at $0$

Consider the function $$ g_a (x) = \begin{cases} x^a\sin ({1\over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ I am looking to determine for which $a$ the map $g_a$ is differentiable on ...
0
votes
1answer
24 views

Differentiating $f(z)=az^2+b\bar zz+c\bar z^2$

Suppose $f(z)=az^2+b\bar zz+c\bar z^2,$ where $a,b,c \in \mathbb C$ are fixed. By differentiating $f(z)$, show that f is complex differentiable at $z$ if and only if $bz+2c\bar z=0.$ So far I've ...
0
votes
1answer
39 views

Differentiating a product symbol

Can someone explain how to differentiate something like $$\prod\limits_{i<j}^N {(x_i-x_j)}$$ with respect to $x_i$ The product starts from 1 and goes to N. I started off by ignoring the $x_j$ as ...
1
vote
1answer
48 views

Properties of a derivative of a function $f(x)$ expressed as a function of $f$

Consider a differentiable function $f(x)$ and let $u(f(x)) = f'(x)$. Normally we solve for $u$ as a function of $x$, but we can also express it as a function of $f$. Some examples: If $f(x) = ...
2
votes
2answers
99 views

How to take the Limits of Logs

How would you take the limit of $$\frac{\log(n!)}{\log(n^n)}$$ as $n\rightarrow\infty$. I believe you have to remove the log raising it to their base. Is this correct ? Thanks.
2
votes
3answers
177 views

Why is integration the inverse of differentiation

Why is integration the inverse of differentiation, I mean why do I get the same function when I integrate and then differentiate the result?
0
votes
2answers
290 views

Finding the derivate of a function using first principles

I want to solve an equation from first principles. The first principles equation is: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ $$f(x) = \frac{1}{\sqrt{x}} \text{ at } x= 1$$ Basically, I ...
0
votes
3answers
62 views

Finding the derivative of a definite integral

$$ G(x)=\int_1^{x^2}(x-t)\sin^2(t)dt $$ Find $ G'(x) $ given $G(x)$. Normally I can solve these types of problems, but I'm thrown off by the two variables present, both $x$ and $t$ under the ...
0
votes
1answer
134 views

What does it mean “The tangent line to the curve $y = f(x)$”

My book says the following: "The tangent line to the curve $y = f(x)$ at the point $P(a,f(a))$ is the line through $P$ with slope" I understand it except for when it says that "$y = f(x)$". I thought ...
0
votes
1answer
47 views

Check if there exists derivative

Let $f: R^2 \rightarrow R$ so that: $$f(x,y)= \begin{cases}\frac{2xy}{x^2+y^2} (x,y) \ne (0,0)\\ 0 \ \ \ \ \ \ \ (x,y)= (0,0)\end{cases}$$ Check if there exists: $ \frac{\partial^2f}{\partial ...
2
votes
1answer
84 views

Find the nature of $f'(x)$ [closed]

Let $f: (-1,\infty) \rightarrow \mathbb{R}$ have a continuous derivative. If $f$ satisfies 1. $f(0) = 1$ 2. $f′(0) = 0$, 3. $(1 + f(x)) f′′(x) = 1 + x$ Then for $x>0,$ $f′(x)$ is: (A) always ...
4
votes
1answer
25 views

Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the ...
1
vote
0answers
35 views

Quotient rule for the Jacobian

Is there an analog to the quotient rule that can be applied to the calculation of the Jacobian? Example: Can the jacobian of a quotient of two functions be decomposed into some series of linear ...
3
votes
0answers
50 views

Wanted: simple invertible function with specified derivative properties

I'm looking for a positive function $F(x)$, defined for positive real numbers, with the following properties. $F(x)$ is expressible with the standard computer math library routines; $F(x)$ is ...
3
votes
2answers
149 views

Gateaux and Frechet derivatives and related notions

Let $X$ and $Y$ be normed real vector spaces, and $f : X \to Y$ a map. Let's say that: G) $f$ is Gateaux differentiable at $x_0 \in X$ if for all directions $v \in X$ the limit $f'(x_0)(v) := ...
1
vote
1answer
19 views

How do you calculate angle between $\frac{x}{5}+\frac{y}{3}=1$ and $x=-1$?

How do you calculate angle between $\frac{x}{5}+\frac{y}{3}=1$ and $x=-1$? What I did: I rewrote $\frac{x}{5}+\frac{y}{3}=1$ as $y=3-\frac{3x}{5}$, therefore $m_{1}$ is $-\frac{3}{5}$, but what is ...
0
votes
3answers
105 views

Having problems with differentiating $y=e^x/x$

Is it a good idea for me to use the quotient rule in this scenario? How should I go about differentiating this problem?
2
votes
1answer
39 views

How to determine quantity of concavity?

Given a function $f(x)$ I can determine whether its concave up or concave down by using the second derivative as it says e. g. here. $$f''(x) > 0 \qquad \text{concave up}$$ $$f''(x) < 0 \qquad ...
2
votes
1answer
86 views

Derivative of a Linear Map

I'm devastatingly incompetent at linear algebra and multivariable calculus. I just cannot understand it at all. Here's the easiest problem from my homework, and my attempt at solving it, and where I ...
1
vote
0answers
90 views

Is this a correct way to prove what the derivative of a polynomial function is?

After trying a polynomial long division problem with a lot of wondering how to go about answering it I proceeded by most likely overcomplicating things but the equation derived seems to work at ...
2
votes
1answer
185 views

Derivative of double summation and dot notation?

I am trying to differentiate the following summation: $$ L(\mu, \tau_1, \ldots, \tau_i)= \sum_{i=1}^v \sum_{t=1}^{r_i} (y_{it}-\mu - \tau_i)^2 $$ $$\frac{dL}{d\mu} = y_{\cdot\cdot}-n\mu - ...
6
votes
1answer
169 views

$\alpha$-derivative (concept)

I found the following definition: Given an real number $\alpha$, we say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is $\alpha$-differentiable at $0$ if exists the limit: $$\lim_{t \to 0^+} ...
0
votes
1answer
39 views

Show there is no solution…

Show that there is no solution to $(\bf D_n − I)p = 0$ except $\bf p = 0$; where $\bf D_n$ is the matrix representing the (first) derivative for degree $n$ polynomials and $\bf p=[c_0; c_1; c_2]$ ...
0
votes
1answer
60 views

Cauchy Riemann and Differentiability

Consider the following proposition. Proposition Let the function $$ f( z ) := u(x,y) + iv(x,y) $$ where $ z = x+iy $ be defined throughout in a $ \eta $-neighbourhood of $ c = a + ib $. Suppose ...
-1
votes
2answers
69 views
0
votes
1answer
33 views

Question id a derivative on a Hilbert space

On a Hilbert space $H$; i have this function: $\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$ where $x_0\in H, p\in C^2([0,\infty),\mathbb{R}),f\in C^2(H,\mathbb{R})$ i want to caculate $\tilde{f}', ...
1
vote
3answers
60 views

if $f(x) = \int_{t=1}^{t=x^2} t\sin^2(t)\operatorname d\!t$ then $\frac{\operatorname d\!f(x)}{\operatorname d\!x}=?$ [duplicate]

$$f(x) = \int \limits_{t=1}^{t=x^2} t\sin^2(t)\operatorname d\!t$$ Do I use U-substitution and have the answer as $$f'(x) = 2x*x^2\sin^2(x^2)$$ Or does this question require integration by parts? ...
0
votes
2answers
43 views

Constructing function tangent to $h(x)$

How do you construct a function $T(x;a)$ for the tangent line to the curve $(sin(4x)+2)^{cos(e^x)} - 1.25$ at the point $x=a=2.2$. Also shown steps would be much appreciated as I don't want just the ...
1
vote
2answers
35 views

Prove that the average of $D_wD_wf(x_0,y_0)$ over all unit vectors $w$ is equal to $\frac{1}{2} \Delta f(x_0,y_0)$ for any smooth function $f$.

Here is a challenge problem from my math professor: Let $w$ be a unit vector in $\mathbb{R}^2$, and let $D_w$ denote the directional derivative with respect to $w$. Prove that for any smooth ...
0
votes
2answers
1k views

How can you find the x-coordinate of the inflection point of the graphs of f'(x) and f''(x)?

So I understand how to find the inflection points for the graph of f(x). But basically, I have been shown a graph of an example function f(x) and asked the state the inflection points of the graph. ...
2
votes
2answers
76 views

Derivative on Hilbert space

Please, on a Hilbert space what is the derivative of $\displaystyle\frac{x}{||x||}$ ? I know that it's equal to $\displaystyle \frac{1}{||x||}-\frac{\langle x,\cdot\rangle}{||x||^3} x$ but can I ...
1
vote
2answers
74 views

Counterexample - Increasing function

My intuition says that the statement is false. Anyone out there know of counterexamples? Suppose $f: R\to R$ and $c\in R$ such that $f'(c) > 0$. So, $\exists \varepsilon> 0 $ such that ...
0
votes
1answer
25 views

Chain rule with inverse function

In a proof, my professor shows: $ s = g^{-1}(u) $ $ ds = \frac{dg^{-1}(u)}{du} du $ , by the chain rule If I were to apply the chain rule to calculate ds, I would not get the du in the denominator. ...
2
votes
1answer
45 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
5
votes
3answers
194 views

$n$th derivative of $e^x \sin x$

Can someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - ...
0
votes
0answers
24 views

Does a point picked for calculating the slope of a tangent line matters?

What I learned is that the slope of tangent is more or less equal to the slope of secant line, and the limit of the secant line is the exact slope of the tangent line. So, do I need to be careful ...
2
votes
1answer
58 views

Simple optimization of cylindrical radius for volume

I'm having trouble solving this simple optimization problem, can't work out where I'm going wrong. A brewery wants to make a cylindrical aluminium beer can which will hold 375ml. (This means the ...
2
votes
3answers
110 views

Finding maximum value for a function

I was working on this question to find the following function's maximum value.Let $$y=f(x)={{(\sqrt{-3+4x-x^2}+4)}}^2 + (x-5)^2$$ where $$1 \le x \le 3$$.I have to find it's maximum value. I tried by ...
2
votes
2answers
34 views

Finding conditions for a with given condition for critical points

$f(x)=\sin2x-8(a+1)\sin x+(4a^2+8a-14$)$x$. $x$ increases for all $x \in \mathbb{R}$ and has no critical points. Find values of $a$. My try: $f'(x)=4(\cos^2x-2(a+1)\cos x+a^2+2a-4)=0$ and ...
9
votes
2answers
157 views

Why are fractal curves nowhere differentiable?

I am a highschool student who stumbled upon fractals when doing a math project. In my research about fractals, I have found that they are nowhere differentiable. Can someone explain this in simple ...
0
votes
1answer
58 views

What is the Frechet derivative of $(u^+)^q$?

I know that if we define $E[u]=\int_\Omega u^+dx$, where $\Omega$ is compact in $R^n$ and $u\in H_0^1(\Omega)$, $u^+:=\max\{u,0\}$, then $E[u]$ is not Frechet differentiable. However, if now I define ...
0
votes
1answer
97 views

proving the quotient rule for derivatives

I have to show the Quotient Rule for derivatives by using just the Product rule and Chain rule. I dont have a clue how to do that. Maybe someone provide me with information. THX
1
vote
0answers
33 views

Legendre transform for HJB PDE

I am trying to understand section 2.3 of the following article: http://www.princeton.edu/~sircar/Public/ARTICLES/montreal.pdf . I think I understand that $H_v(t,g(t,z)) =z$, because by definition of ...
1
vote
1answer
142 views

Taylor series and 100th derivative on this function

I have this real function: $f(x) = \frac{1}{(x^2-2x+3)^2}$ and I need to find Taylor series at $x = 1$ and find 100th derivative at $f^{(100)}(1)$. Can anybody help me???
1
vote
0answers
29 views

How can I find the Min and max of this question?

I have been trying for the past 2 hours on this question and cannot seem to figure out the answer. So far I have gotten the 'green' bits correct. Someone Help please
3
votes
3answers
71 views

What is the term for whatever is being differentiated?

When we integrate a function: $ \int^b_a {2\over x^2} dx$ The expression to be integrated (is this case $ {2\over x^2} $) is referred to as the integrand. When we differentiate a function: $ {d ...
3
votes
2answers
109 views

Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$ \rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...