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

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0
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3answers
73 views

Prove the derivative of $\sin(1/x)$ exists

How do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...
0
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1answer
43 views

Is a constant a $ C^\infty $ function?

Is the function $f(x)= t $ (where t is a constant) a $ C^\infty $ function? The derivative would be zero, and it is continous everywhere. But still a classmate of mine doesn't seem to be convinced ...
5
votes
2answers
63 views

Rudin's chain rule: Why is continuity at interval necessary?

Theorem 5.5, Rudin's Principles of Mathematical analysis says: Suppose $f$ is continuous on $\color{red}{[a,b]}$,$ f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which ...
0
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1answer
62 views

How can I solve this equation [closed]

How can I solve this differential equation : $$2x\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3+\frac{dy}{dx} =0$$
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1answer
24 views

What properties do I have if I know $f$ and $f^{-1}$inverse are differentiable?

My goal is to show that $(f^{-1})'(y) = 1/[f'(f^{-1}(y)]$ for all $y$ in $(a,b)$. I have no idea where to start. I know that $f^{-1}$ and $f$ are differentiable.
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2answers
42 views

complex differentiation, alternative way?

$\partial/\partial\bar{z}$ is defined as $1/2[\partial/\partial x+i\partial/\partial y]$. So lets say you have a function $f(z,\bar{z})$ in order to find $\partial f/\partial \bar{z}$ I have to write ...
1
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2answers
51 views

Laplacian of a radial function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a radial function, i.e. $f(x)=f(r)$ with $r:=\left\|x\right\|_2$. As stated at Wikipedia $$\Delta f=\frac{1}{r^{n-1}}\frac{d}{dr}(r^{n-1}f')$$ What's the most ...
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1answer
20 views

Differentiation of composed function.

Using the rule of differentiation composed function calculate the first order partial derivatives of $x$ and $y$: $$ z = f(u,v,w) = \arcsin \frac{u}{v+w}$$ $$u= e^\frac{x}{y}, v= x^2 + y^2, w =2xy$$ ...
4
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1answer
42 views

convolution with $C^{\infty}$ produces $C^{\infty}$

Problem: So I have the following function in ...
-3
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2answers
54 views

Why does differentiability implies continuity, but continuity does not implies differentiability?

Why does differentiability implies continuity, but continuity does not implies differentiability? I am more interested in the part about a continuous function not being differentiable. Well, all ...
0
votes
1answer
24 views

Limit of Derivative and Derivative of Limit

Assuming the integral is finite and $f$ is continuous, does this argument always work? If not, what do we need more? $\displaystyle \frac{d}{dx}\int_x^{\infty} f(t) \, \mathrm{d}t =$ $\displaystyle ...
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2answers
15 views

Condition for a line to be tangent of a parabola

I just read that the line y = kx + n will be tangent line of a parabola y^2 = 2px if derivatives of both of them are the same. ...
0
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3answers
56 views

Find the derivative of y = $\sqrt{xe^{2x} + 3e^{-x^2}}$

I am trying to find the derivative of this problem but I am not sure where to start. Any help is appreciated. Find the derivative of $$y = \sqrt{xe^{2x} + 3e^{-x^2}}$$
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vote
0answers
14 views

Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
1
vote
1answer
39 views

Differentiability of monotonic functions

If a function is monotonic on set E. Is f differentiable almost everywhere? I have proved for case E closed bounded or open intervals, hence all open sets. But in general I am not able to figure it ...
0
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0answers
17 views

derivative of a specific integral

I would like to get the derivative of the function $f(z)$ defined as: $$ f(z)=\int_{z+dz}^{z}k\left(h(z')-a\right).g(z')dz' $$ where $a$ and $k$ are constant and $h(z)$ and $g(z)$ functions. How do I ...
0
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2answers
36 views

How to take the derivative with respect to $x^0$? [closed]

I need to take second derivative of $$\sin(\theta \cdot x^0 -y_1 \cdot x^1 -y_2 \cdot x^2),\theta>0, y_1,y_2 \in \mathbb{R}$$ with respect to $x^0$ which is equal to $1$. What to do?
6
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2answers
72 views

How to solve $\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$

Could you help me to prove $$\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$$ where ${\rm B}(a,b)$ is Beta function.
0
votes
1answer
60 views

Where is the slope of the tangent to an astroid equal to -1?

The equation of astroid is $x^{2/3} + y^{2/3} = a^{2/3}$. Find the points where the slope of the tangent to the astroid is equal to $-1$. I got the derivative to be $-y^{1/3}/x^{1/3}$ and so I ...
0
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2answers
29 views

Limit definition of derivative

How do I go about doing this question? Am i using the right formula?
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1answer
29 views

Can definite integral be of the form $\int_{h(x)}^{g(x)}F'(x)dx exist$?

In theory, I can write down an integral of the form $$I(x)=\int_{h(x)}^{g(x)}F'(x)dx$$ and solve it as $$I(x)=F(g(x))-F(h(x))$$ Out of curiosity, I plugged this into my calculator and was given a ...
1
vote
1answer
25 views

Derivative of integral with x as the lower limit

Question: Let $$F(x) =\int_{x^3}^{5}(cos^2t-te^t)dt $$ Find $F'(x)$ We were not explicitly taught about this during the semester but from what I can gather from online readings is that ...
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2answers
48 views

Show that $F$ and $G$ differ by a constant

Suppose $F$ and $G$ are differentiable functions defined on $[a,b]$ such that $F'(x)=G'(x)$ for all $x\in[a,b]$. Using the fundamental theorem of calculus, show that $F$ and $G$ differ by a constant. ...
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0answers
8 views

differentiating with respect to vectors.

if a function f has domain R^d (column vectors) and codomain R (numbers), then its derivative has domain R^d (column vectors) and codomain R^(1xn) (row vectors). What is the codomain of the 2nd ...
0
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1answer
36 views

Differentiating a vector valued function

If I have a function $y(x)=f(a+x(b-a))$ where $a, b$ are constant vectors, and $y: \mathbb{R} \rightarrow \mathbb{R}$, what would $\frac{dy}{dx}$ be in terms of $f$? I know the chain rule would be ...
0
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1answer
37 views

Further Trigonometric Differentiation [closed]

I need to derive the maximum point of this equation for a modelling problem but am not too sure if my differentiated value is accurate. Would appreciate if someone could give it a shot! ...
4
votes
1answer
79 views

Computing an explicit Radon-Nikodym derivative

Q/ let $\lambda$ be the Lebesgue measure and $\delta_0$ be the Dirac measure at 0. Show that $\lambda$ is abs cts wrt $\lambda+\delta_0$ (have done this part) and find the R-N derivative ...
0
votes
1answer
59 views

How to find the $k$th derivative of $1/x^y$ with respect to $x$?

What would be the solution to the $k^{th}$ derivative of the following function $$\dfrac{1}{x^y}$$ With respect to $x$ where y is a constant. I have calculated the first derivative ...
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2answers
40 views

[edited]Prove that $f(x)=0$ exists in a certain interval.

I have $f:R \rightarrow R$, $f(0)=-1$ and $f'(x) \ge1$ $\forall x$. I need to show that $f(x)=0$, for some $x\in[0,1]$ I know that I need to use mean value theorem and intermediate value theorem. ...
0
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2answers
40 views

Can Rolle's Theorem be true for the critical point where derivative doesnt exist?

there is the problem that I met At 0 the derivative of f(x) doesn't exist so 0 is the critical number but the conclusion of Rolle's theorem is the f'(c) (here c=0) must be 0. Are there any ...
2
votes
1answer
32 views

limit of function by using derivative

Let $f: (0,\infty) \to \mathbb R$ be a differentiable function such that $f^{\prime}(x)= \frac{x^2 - (f(x))^2}{x^2((f(x))^2+1)}$. Prove that $$\lim_{x \to \infty}f(x)=\infty$$
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3answers
54 views

Finding the derivative of $\frac1{\sqrt{x^2-1}}$

Use first principles to find the derivative of the following. $$\frac1{\sqrt{x^2-1}}$$
1
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1answer
32 views

Show that the tangent only touches the graph in one point.

Let $f: \mathbb R\to \mathbb R$ be such that $f'$ is increasing. Show that for all $x$ the tangent line through the point $(x, f(x))$ only touches the graph in that point. So I'm kinda stuck with ...
0
votes
2answers
63 views

Limit of this integral

$$\lim_{x\to0}\frac{\int_x^{x^2}\sinh(t)\sin(t)\,dt}{\int_0^x t^3\csc(t)\,dt}.$$ I'm not sure what to do for this I tried integrating both the numerator and denominator separately but I wasn't ...
3
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6answers
142 views

Prove that $\frac{d(\log(x))}{dx}=\frac{1}{x}$

Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it? I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} ...
1
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1answer
18 views

Prove the following equality regarding partial derivatives

Let $f:\Omega\subset\mathbb{R^2\to\mathbb{R}}$ be a function such that $f\in\mathit{C^1}(\Omega)$. Now, consider the function: $$g(x,y,z):=x^4f(y/x,z/x)$$ Prove that $$x\frac{\partial g}{\partial ...
0
votes
0answers
26 views

How to check whether a linear map on integral domains is a formal derivative

I have an elementary question on formal derivatives. Assume $A=K[X,Y,Z]/I$ is an integral domain (for example $I$ is a prime ideal and K is the field of rationals). Let $d:A\to A$ be a linear map. Is ...
1
vote
1answer
46 views

Is this function differentiable in $(0,0)$

Consider the function: $$f:\mathbb{R^2}\rightarrow\mathbb{R}$$ $$f(x,y)=\frac{x^2y^2}{x^4+y^2}\forall (x,y)\neq(0,0)$$ $$f(0,0)=0$$ It's clearly differentiable for all $(x,y)\neq(0,0)$. I have shown ...
12
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1answer
586 views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
0
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0answers
31 views

Functions differentiable at the irrationals and not differentiable elsewhere

I provide here an example of a real function that is differentiable at all reals except at $0$ and which has a bounded derivative. Edit: and which do not have left and right derivatives at $0$. Do ...
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2answers
35 views

Using the Definition of derivative to derive the derivative of a function

Assume $f$ is differentiable at point $a$, and $f(a)>0$. Determine the derivative of $$g(x)=x\sqrt{f(x)}$$ in terms of $f'(a)$. What I have done is substituting in whatever is given and I don't ...
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5answers
91 views

I have proven that $e^x > x^3$ for $x>5$, can I prove that $\lim \frac{x^3}{e^x} = 0$?

In order to calculate the limit $$\lim_{x\to\infty} \frac{x^3}{e^x} = 0$$ I've verified that: $$f(x) = e^x-x^3\\f'(x) = e^x-3x^2\\f''(x) = e^x-6x\\f'''(x) = e^x-6$$ Note that $x>3 \implies ...
3
votes
3answers
60 views

Find $a$ such that $x^3 +3x^2-9x+a = 0$ has only one real root

I have the function $$x^3 +3x^2-9x+a$$ If I take the derivative, I have $$3x^2+6x-9$$ This is a parabola with a negative part. So my function isn't always increasing, and therefore can have more ...
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0answers
16 views

Possibly notation problems involving Integration and pullbacks on k-forms

$^*$ means the pullback of a k-form in this example. I cannot see how the underlined expressions have been found 1) I think that $(c \circ G)^*\omega = G^*(c^* \omega)$ but I cannot see why $c^* ...
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3answers
51 views

How can I differentiate this equation? $y = \sqrt[4]{\frac{(x^3+2\sqrt{x})^2(x-sinx)^5}{(e^{-2x}+3x)^3}}$

$y = \sqrt[4]{\frac{(x^3+2\sqrt{x})^2(x-sinx)^5}{(e^{-2x}+3x)^3}}$ I tried removing the root but that got me no where
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0answers
26 views

Gateaux and Frechet differentiability

Please help me to investigate Gateaux and Frechet differentiability of the functional $x \rightarrow ||x||_c$ depending on $x \in c$. The same about functionals $x \rightarrow ||x||_{c_0},\ x \in c_0$ ...
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3answers
73 views

Prove that $x^3 -3x^2 +6 = 0$ has only one real root

I know that if I take the derivative of $$x^3 -3x^2 +6 = 0$$ and prove it is always greater than zero, I'll find that this functions is always increasing, and therefore if I find an interval where ...
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1answer
40 views

Showing that two equations are equal using chain rule.

Let $u = f(x,y)$, with $x= r \cos\theta$, $y =r\sin\theta$. Show that $$\left(\frac{\partial u}{\partial r}\right)^2+\frac{1}{r^2}\left(\frac{\partial u}{\partial ...
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vote
3answers
69 views

Find $\lim_{x \rightarrow 0} (\frac{\tan x}{x})^{x^{-2}}$

I see that this is in the $1^ \infty$ form, so I've taken log to get: $\lim_{x \rightarrow 0} \log( \frac{\tan x}{x})^{\frac{1}{x^2}}$ which is equivalent to $\lim_{x \rightarrow 0} \frac{\log ( ...
1
vote
0answers
36 views

Derivative for eigenvalue with respect to 1st / 2nd / 3rd invariant of a matrix

Definition There is a 3 by 3 matrix $A$ where $Ax=\lambda x$, so the $\lambda$, where $\lambda$ and $x$ are eigenvalues and eigenvectors of matrix $A$. And then we have the invariants of the matrix, ...