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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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
votes
2answers
70 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?
1
vote
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 ...
1
vote
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. ...
0
votes
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
votes
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
78 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 ...
1
vote
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 $$\lim_{x \to \infty}f(x)=\infty$$
1
vote
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
vote
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
votes
6answers
141 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
vote
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
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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
votes
1answer
584 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
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
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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^* ...
1
vote
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
1
vote
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 $x \rightarrow ||x||_{c_0},\ x \in c_0$ ...
1
vote
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 ...
0
votes
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 ...
1
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, ...
0
votes
0answers
25 views

The idea behind the Leibniz notation and the use of it in integral.

I am having trouble in understanding why the substitution rule in integral or the chain rule work. Mostly, it is because I cannot understand the idea of Leibniz notation. In my book it is defined as ...
0
votes
0answers
21 views

Finding a generalized form for taking the n$^{th}$ derivative of a falling factorial

I would like to make $$ \frac{d^n}{dx^n}[(x)_c] = n! \times e_{c-n}(x,x-1,\cdots,x-c+1) $$ Where $e_{c-n}(x,x-1,x-2,\cdots,x-c+1)$ is the elementary symmetric polynomial function But lets say that ...
1
vote
1answer
63 views

Find the $n^{th}$ order derivative of $x^n \ln x$

I'm doing it completely wrong, I'm sure, but I'll still show my attempt: $n^\text{th}$ order derivative of $x^n$ is $n!$ and of $\ln x$ is $(-1)^{(n-1)} (n-1)! x^{-n}$ So, using Leibnitz rule I got ...
0
votes
1answer
14 views

Verification of proof regarding limit and derivative at infinity

Ok so I have been working through Calculus by Spivak and stumbled upon a theorem which I found hard to prove ,and solution in answer book seems to be wrong.So I need you to help me verify my proof. ...
2
votes
1answer
34 views

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ smooth, $ g(x,y)= x^3 + y^3$ and $g \circ f \equiv 0$, then $\det Df \equiv 0$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function and $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $(x,y) \mapsto x^3 + y^3$. Assume that $g \circ f$ is identically $0$. ...
0
votes
0answers
24 views

Second derivative relating to continuity

Given a function $f$ such that the integral $A(x)=\int_a^x{f(t)dt}$ exists in an interval $[a,b]$. Let $c$ be a point in the open interval $(a,b)$. Consider the following ten statements about this $f$ ...
0
votes
1answer
34 views

$f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$

Assume that $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$ for all $x\neq 0$ and $\alpha > 0$. Then I want to prove that $x \cdot ...
1
vote
1answer
49 views

If $f: U \rightarrow \mathbb{R}^n$ differentiable such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$, then $\det \mathbf{J}_f(x) \neq 0$

Let $f: U \rightarrow \mathbb{R}^n$ be a differentiable function on an open subset $U$ of $\mathbb{R}^n$. Assume that there exists $c>0$ such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$. ...
19
votes
2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
0
votes
3answers
72 views

n-th order derivative of a function

Find the n-th derivative of the following functions: $y = x\sqrt{1+x^2}$ $y = \dfrac{x}{\sqrt{x-x^2}}$ All help will be appreciated. Thank you!
2
votes
1answer
32 views

Directional derivative of a function

Feel like I may have gone wrong somewhere with this question: Find the directional derivative of the function $f(x,y) = \displaystyle\dfrac{2x}{x-y}$ at the point $P(1, 0)$ in the direction of the ...
3
votes
2answers
91 views

Whats the derivative of $\sqrt{4+|x|}$ using first principle

Here is my attempt: $$f(x)=\sqrt{4+|x|}$$ $$f`(x) = \lim_{h\to0} \frac{\sqrt{4+|x-h|}-\sqrt{4+|x|}}{h}$$ multiplying by the conjugate: $$\lim_{h\to0} ...
1
vote
1answer
31 views

Differentiation of multivariate function with respect to another multivariate function [closed]

How do I calculate the derivative $\frac{df(x,y)}{d(xy)}$, given that $x,y\neq 0$ and assuming that the derivative exists?
0
votes
1answer
23 views

Point of inflection and third derivative

In my textbook there is a confusing statement. If $f'''(ξ)=0 $ and $f''(ξ)\ne0$ then $ξ$ is inflection point. However this confuses me as it is contrary to book example and this. Also in class notes ...
0
votes
1answer
16 views

Differentiating a function that includes vectors using the chain rule

I am trying to differentiate the function: $$g(x) = f(3\vec k + x(\vec l + \vec k))$$ where $\vec k$ and $\vec l$ are in $\mathbb R^n$ and $x$ is in $\mathbb R$. I think I need to use the chain ...
0
votes
1answer
44 views

Evaluate the integral, and then take the derivative of it.

I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer. So I first evaluated the top section And when I did that I got ...