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

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2
votes
2answers
50 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
16
votes
2answers
489 views

What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so ...
1
vote
0answers
5 views

Discretization of v*(du/dx)

I am trying to discretize the term: $$\underline{v}\frac{d\underline{u}}{dx}$$ using finite differences or evaluate $$\int_{\Gamma}\underline{v}\frac{d\underline{u}}{dx}.\underline{n}d\Gamma$$ ...
2
votes
1answer
40 views

If $f$ has derivative at $1$ and $\lim_{h \to 0} {\frac{f(1+h)}{h} }=1$, then $f'(1)=0=f(1)$

I need to prove that if $f(x)$ has derivative at $x=1$ and if $\lim_{h \to 0} {\frac{f(1+h)}{h} }=1$. then I need to prove that $f'(1)=0$ $f(1)=0$. It's pretty obovious if using arithmetic of limits, ...
0
votes
1answer
31 views

Continuity and differentiablity [on hold]

True or False ? If $f : \mathbb R \to \mathbb R$ satisfies $$|f(x) − f(y)| ≤ |x − y|^{\sqrt{2}}$$ for all $x, y \in R$, then $f$ must be a constant function. Let $f : \mathbb R\to \mathbb R$ be ...
0
votes
2answers
34 views

The derivative of $z=x^2+xy+ y^2$

I have got confused about this problem, what I have thought was differentiating this with respect to $x$ gives - $\frac{dz}{dx} = 2x + x \frac{dy}{dx} + y + 2y \frac{dy}{dx}$ But, I came across an ...
3
votes
2answers
113 views

Question about left and right derivative.

Let $f(x):\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $\forall x\in\mathbb{R}$ there exist: $$f'_+(x)=\lim_{\delta\rightarrow 0^+}\frac{f(x+\delta)-f(x)}{\delta}$$ ...
1
vote
1answer
18 views

Issue differentiating the Lambert W function

I want to differentiate the Lambert W function (the inverse of $y = xe^x$), I didn't think it would be that difficult a problem but it's causing me some problems. I tried this method: (1.) Implicitly ...
2
votes
2answers
3k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
4
votes
4answers
189 views

Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
3
votes
1answer
63 views

Computing the derivative of a transformation matrix

I am trying to find a geometric transformation between two images, where the transformation is a simple scaling matrix. So, if I denote the two image functions as $r$ and $f$ and the scaling matrix as ...
1
vote
2answers
30 views

Definition of reciprocal derivative

Suppose I define $y(x)=x^3$. Then, $$\frac{\mathrm dy(x)}{\mathrm dx}=3x^2.$$ however I don't see how $\displaystyle \frac{\mathrm dx(y)}{\mathrm dy}$=$\frac{1}{3x^2}$ because I never explicitly ...
-1
votes
4answers
74 views

If $x^2 +xy =10$ then when $x=2$ what is $\frac{\mathrm dy}{\mathrm dx}$?

If $x^2 +xy =10$ then when $x=2$ what is $\frac{\mathrm dy}{\mathrm dx}$? I solved for $y=3$ before I did the product rule and i'm not sure if that was the correct way to approach it.
1
vote
3answers
121 views

Finding $\frac{\mathrm d}{\mathrm dx}(6)$ by first principles

I had an exercise in my text book: Find $\frac{\mathrm d}{\mathrm dx}(6)$ by first principles. The answer that they gave was as follows: $$\lim_{h\to 0} \frac{6-6}{h} = 0$$ However surely that ...
0
votes
3answers
35 views

real valued functions

Let $F(x,y)=H(xy)$ be a real valued function defined over non-negative reals. $H(\cdot)$ is differentiable in its argument. Take partial derivatives $F_x=yH_x(xy)$ and $F_y=xH_y(xy)$. Can we say that ...
-5
votes
1answer
49 views

How to show that a given function is a solution of differential equation?

I have been trying to prove this for awhile but in any way that I try it doesn't give me the same required answer that I must show, any ideas? If ${y =\sqrt{x} + \dfrac{1}{\sqrt{x}}}$, ...
2
votes
1answer
44 views

Mean Value Theorem - End point question

So I'm beginning numerical analysis and an interesting thing was brought up in class. I know the rules for MVT are: F is continuous on [a,b] F is differentiable on (a,b) So a question was brought ...
0
votes
1answer
25 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
1
vote
1answer
14 views

contraction identity on $k$-forms

$i_\mathbb{X} \omega $ is the contraction of $\omega$ with respect to $\mathbb{X}$. In my notes it is stated that $i_\hat{\mathbb{X}} dx = dx(\hat{\mathbb{X_t}})$. I cannot see how this fits the ...
0
votes
0answers
32 views

Estimate for $f^2$ on a Ball from below

Let $f\in C^\infty(B_R(0))$, where $R>0$. For $0<\sigma<1$ require the following properties on $f$: $$ 0\leq f\leq 1,\ f=1 \text{ on } B_{(1-\sigma)R},\ f=0 \text { on }\partial B_R,\ ...
-2
votes
1answer
101 views

Calculus (derivatives/slopes) [on hold]

Consider the equation $x^2+3y^2+xy=3$. a) Write an expression for the slope of the curve at any point (x,y). The answer is: $dy/dx = (-2x-y)/(6y+x)$ b) Find the equation of the normal line ...
0
votes
4answers
56 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
2
votes
5answers
51 views

If angular velocity $\omega=\sqrt{\frac{3g\sin\theta}{2a}}$ can I find angular acceleration $\alpha$ by differentiating $\omega$?

It was my understanding that angular acceleration is the derivative of angular velocity. The reason I ask is Thanks.
2
votes
5answers
99 views

Showing the $n$-th derivative of $\cos x$ by induction

I was asked to show that the $n$-th derivative of $\cos x$ is $\cos(\frac{n\pi}{2} + x)$. My progress : By induction, I proved it was true for $n=1$. Then I assumed it was true for $n = k$ so now I ...
4
votes
1answer
64 views

Prove that there exist $a \in [-1,1]$, such that $f'''(a)=3f(1)-3f(-1)-6f'(0)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function. Prove that there exists $a \in [-1,1]$ such that $$ f'''(a) = 3f(1)-3f(-1)-6f'(0)$$ Any hint/idea's how to ...
0
votes
0answers
7 views

Simplifying general formula for fractional derivative by removed derivative of integral.

On the wikipage about fractional calculus, there's a general formula for the fractional derivative: $D^\alpha$ is the derivative operator. $$D^\alpha ...
0
votes
1answer
10 views

Equivalence relation of differential forms

My notes claim that $\displaystyle d\omega (x) = \frac{1}{k!} d\omega_{i_1\cdots i_k} \wedge f^{(i_1)}\wedge\cdots\wedge f^{(i_k)}$ is equivalent to $\displaystyle d\omega(x) = \frac{1}{k!} ...
1
vote
0answers
91 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
2
votes
2answers
32 views

How to find local maximum of the function $f(x) = x^3-9x^2+24x+4$?

Give the value of x where the function $f(x) = x^3-9x^2+24x+4$ has a local maximum. a) -4 b) 4 c) 2 d) 3 e) -2 I graphed it and I'm not sure how to find the local max
3
votes
1answer
34 views

Studying the function $f(x) = x^4-6x^2$ using derivatives: minima, maxima, inflection, concavity

(I know this is my second question today, but I'm explaining what I'm doing so I hope it's okay) Consider the graph of $f(x) = x^4-6x^2$. a) Find the relative maxima and minima (both x and y ...
1
vote
5answers
63 views

Derivative of $\ln(1+\sin 2x)$

Differentiate the equation $y=\ln(1+\sin2x)$. It will be something to do with the $\frac{d}{dx}\{\ln\:x\}=\frac{1}{x}$ rule, but I'm not sure how to deal with the $\sin2x$ term.
0
votes
3answers
50 views

Differentiate a recurrence relation

How do I calculate the derivative of an equation like: $z_n = (z_{n-1} + c)^2$ with respect to $n$ where $z_0 = 0$ and $z,c \in \mathbb{C}$ I suspect that, for a given $z$, the derivative is not ...
-2
votes
1answer
53 views

derivability don't imply partial to be continuous ? example

Is $$f(x,y) =\begin{cases} x^2+2x+2y & \text{ for } (x,y)\neq (0,0) \\ y^2 & \text{ for } (x,y)=(0,0) \end{cases}$$ derivable? But its partials are not continuous?
0
votes
2answers
38 views

Differential identity and wedge products

Apparently $dx^{i_1} \wedge ... \wedge dx^{i_k}=d(x^{i_1}dx^{i_2}\wedge ... \wedge dx^{i_k})$ which I cannot see proved anywhere in my notes. It just stated as if it is obvious which I don't believe ...
1
vote
1answer
54 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
1
vote
0answers
23 views

How t find z (unknown) in Runge-Kutta question

I'm trying to solve the below question solve $\dfrac{dx}{dy}=\dfrac{1}{x+y}$ for $x=0.5$ to $z$ using R-K (order $4$) with $x_0=0$, $y_0=1$ (take $h=0.5$). Please help me and tell me how to ...
0
votes
1answer
31 views

continuity, discontinuity derivative and relation to being derivative but its partials are not continuous

is $$f(x) =\begin{cases} x^2\sin(\frac{1}{x}) \mbox{ for } x\neq 0 \\ 0 \mbox{ for } x= 0\end{cases}$$ a continuous function specially at point x=0? And why being derivable its derivative is not ...
0
votes
4answers
149 views

derivative of $x\cdot|\sin x|$

I have the function $f(x)=x|\sin x|$, and I need to see in which points the function has derivatives. I tried to solve it by using the definition of limit but it's complicated. It's pretty obvious ...
2
votes
2answers
63 views

Part of proof that $d^2\omega=0$

The following comes from the proof in differentiable manifolds that $d^2\omega=0$. Let $f$ belong to the set of $0$-forms. From definition I have that $\displaystyle df = \frac{\partial f}{\partial ...
8
votes
2answers
373 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
7
votes
5answers
182 views

Is there an easier way to find $\frac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$ than using the product rule repeatedly?

Find $\dfrac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$. I know how to solve this problem by repeatedly using the product rule, but I was wondering if there is a short cut. Thanks.
2
votes
0answers
20 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
0
votes
1answer
18 views

Minimizing an error function by deriving a system of linear equations

Consider the following formula: $$E(\mathbf{w}) = \frac{1}{2}\sum_{n=1}^{N}\{y(x_n,\mathbf{w})-t_n\}^2$$ where $\mathbf{w}$ is a vector of weights; $x_n$ and $t_n$ come from two vectors of length ...
3
votes
4answers
135 views

What is $\frac{d(\arctan(x))}{dx}$?

Let $v= \arctan{x}$. Now I want to find $\frac{dv}{dx}$. My method is this: Rearranging yields $\tan(v) = x$ and so $dx = \sec^2(v)dv$. How do I simplify from here? Of course I could do something like ...
1
vote
1answer
28 views

Show that the value of $\frac{\text{d}^{2r+1}y}{\text{d}x^{2r+1}}$ when $x=0$ is $\frac{1}{2^{2r}}\left(\frac{(2r)!}{r!}\right)^2$

The question originally asks you to prove that if $y=\sin^{-1}(x)+(\sin^{-1}(x))^2$ that: $(1-x^2)y''-x y'$ is independent of $x$. I get that $(1-x^2)y''-x y'=2$ hence proving the first part. The ...
1
vote
1answer
32 views

Differentiability of non-analytic complex functions

Any complex function that is analytic on an open set is differentiable on that set. But can a function fail to be analytic on an open set but still be differentiable? For example, the function ...
0
votes
3answers
44 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
0
votes
0answers
18 views

Which property can be used to derive a differential equation for a reparametrization

With $0\le t\le1$, two space curves given by: $$c_1(t)=(1,t,0)\quad\quad c_2(t)=(0,t,2t(1-t))$$ One of them, say $c_1$, must be reparametrized by $r(t)$ in order to minimize the area between the ...
0
votes
1answer
320 views

Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
2
votes
2answers
51 views

Properties inherited by $f\circ g$ from $f$

Suppose $f,g:\mathbb{R}\to \mathbb{R}$ Prop: Suppose $g$ and $f \circ g$ are ______, and $g$ achieves every value in $\mathbb{R}$. Then $f$ is ______. If in the blanks we put the word ...