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

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1answer
18 views

Finding derivatives for a Cauchy-Euler ODE

I'm having some trouble following along with the reduction of the Cauchy-Euler equation into a linear one with constant coefficients. I've been trying to follow along with the work here, but I don't ...
1
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1answer
38 views

Trick with differentials from $\frac{dr}{ds} \to \frac{dr}{dt}$

I need to come from $r''=\frac{d^2r}{ds^2}$ to $\ddot{r}=\frac{d^2r}{dt^2}.$ I know that $r' = \frac{\dot{r}}{|\dot{r}|}.$ And more $1/\dot{s} = t' = \frac{1}{\sqrt{\dot{r}\cdot\dot{r}}}.$ You can see ...
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2answers
36 views

What are the critical points of $x-4\sqrt{x+1}$?

A critical point $c$ is defined as $f'(c) = 0$ or $f'(c) = $ undefined. This definition is taken from this video. if $$f(x) = x-4\sqrt{x+1}$$ then $$f'(x) = 1 - \frac{2}{\sqrt{x+1}}$$ To find the ...
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2answers
57 views

Prove that $\nabla_X tr(X^TAX)= (A+A^T)X $ where $A \in \mathbb{C^{m*m}}$ and $X \in \mathbb{R^{m*n}}$

Prove that $\nabla_X tr(X^TAX)= (A+A^T)X $ where $A \in \mathbb{C^{m*m}}$ and $X \in \mathbb{R^{m*n}} $ . 1.) Same proof stands when $ A\in \mathbb{C}$ or $ A\in \mathbb{R}$ ? 2.) What is the ...
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2answers
68 views

How do you find the inflection point of this graph?

The graph is this: $$ \frac{(x+1)^3 - 4(x+1)^2 + 4(x+1)}{(x+1)^2 - 2(x+1) + 1} $$ I know you can find second derivative and then solve for values that make it undefined or 0, but I was told ...
2
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1answer
66 views

When $ \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0$ implies $\lim_{x\rightarrow +\infty}f'(x)=0$?

I have just solved a problem: Let $f:[0,+\infty)\rightarrow \mathbb{R}$ be continuous on $[0,+\infty)$ and differentiable on $(0,+\infty)$. If $\displaystyle \lim_{x\rightarrow +\infty}f'(x)=0$, ...
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1answer
30 views

Proving question (Differentiation)

Given that $y=\tan (x+\frac{\pi}{4})+1$, show that $\frac{d^2y}{dx^2}=2y(\frac{dy}{dx}$). I got $\frac{dy}{dx}=\sec^2 (\frac{\pi}{4}+x)$ but I don't know how to proceed. P/s: The question states ...
2
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1answer
46 views

Prove $f'(0)=0$, if $|f(x)|≤x^2$

Let $f:(-a,a) \longrightarrow R$, $a>0$. Such that $$ |f(x)|≤x^2 $$ What I did was taking out the module bars so I get $-x^2≤f(x)≤x^2$ and I see that at $x=0$ the function must be zero. I see ...
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1answer
20 views

how does this given condition translate into this specific interval (calculus)

in my calculus book i am learning how derivatives affect the shape of the graph, for one problem i am supposed to sketch a graph given the conditions. I understand when f'(x)>0 the f(x) will increase ...
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2answers
77 views

Statement about Rolle's Theorem (true or false?)

There's a statement, that I believe is false Between two distinct zeroes of a polynomial $p$, there is a number $c$ such that $p′(c) = 0$. Here is my reasoning: A polynomial of an even ...
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3answers
85 views

Does $f(x)>g(x)$ imply $\frac{d}{dx}f(x)>\frac{d}{dx}g(x)$?

Is it true that $f(x)>g(x) \implies \frac{d}{dx}f(x)>\frac{d}{dx}g(x)$? What about $|f(x)|>|g(x)| \implies \frac{d}{dx}|f(x)|>\frac{d}{dx}|g(x)|$?
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1answer
30 views

Maximizing trapezoid area

Find the trapezoid of largest area that can be inscribed in the region bound by the graph of $y=4-x^2$, and the $x$-axis. So, I know that you need to maximize the area of the trapezoid, which I used ...
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2answers
43 views

What does the notation $\frac{\partial(x,y)}{\partial(u,v)}$ mean?

Suppose G$(u,v) = (x, y, z)$ In terms of derivatives, what does $\frac{\partial(x,y)}{\partial(u,v)}$ mean?
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2answers
46 views

How to differentiate $y=(x+1)^3/x^{3/2}$ and $y=2x^4/(b^2-x^2)$

I need to solve a list of derivatives to help me on an exam; however, I'm in doubt when they use another variable (constant) or when I have a fraction with functions that use the power rule. For ...
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6answers
274 views

Derivative: $e^x$. [duplicate]

How do you differentiate $e^x$? I looked on many sites, including similar questions here but most answers seemed circular. The only known definition of $e$ to be used in this proof is $$ e=\lim_{n ...
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0answers
30 views

Partial derivatives generalization

Let $F:U\rightarrow\mathbb{R}^m$ be a function for some open $U\subseteq\mathbb{R}^n$. If all partial derivatives of $F$ exist and are continuous in a neighborhood of some point $x_0\in U$, then $F$ ...
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0answers
26 views

Curious coincidence: $ 2x^2 (x^2-1) =3(y^2-1) $ and $ x(x-1)/2 =2^n -1 $

Why is it such a coincidence that for both diophantine equations $ 2x^2 (x^2-1) =3(y^2-1) $ and $ x(x-1)/2 =2^n -1 $ have five positive integer solutions $x=1,2,3,6,91$? Source: example 29
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3answers
1k views

Computing a higher (2015) order partial derivative of $1/(x^2+y^2)$

Suppose $$f(x,y) = \frac{1}{x^2 + y^2}\text{.}$$ Find $$\frac{\partial^{2015} f}{\partial x^{2015}}\text{.}$$
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1answer
39 views

Differentiation / Limit problem from intro real analysis textbook

Assume that $f: (-1,1) \to \mathbb{R}$ and $f'(0)$ exists. If the sequences $\alpha_n, \beta_n \to 0$ as $n \to \infty$, define the difference quotient $$D_n = \frac{f(\beta_n) - f(\alpha_n)}{\beta_n ...
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2answers
43 views

Why only the numerator is derived?

Why the derivative of $y = \frac{x^5}{a+b}-\frac{x^2}{a-b}-x$ is solved by deriving just the numerators? The solution is $\frac{dy}{dx}=\frac{5x^4}{a-b}-\frac{2x}{a-b}-1$.
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1answer
30 views

How to take derivative of algebraic function with respect to s

I am reading some books and papers on operational calculus (which is quite similar to Laplace transform) and am unable to understand some of the workings/derivations. So I guess I need to understand ...
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2answers
23 views

Prove that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses

Prove that if $f$ is a function such that $f'(x) > 0$ $\forall x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
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0answers
18 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
3
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1answer
89 views

How to show that $\frac{-1}{x^2}=0$ has no solutions?

I must check if the function $f(x)=\frac{1}{x}$ has a tangent line with slope $0$. I took the derivative: $$\left[\frac{1}{x}\right]'=\frac{-1}{x^2}$$ And then: $$\frac{-1}{x^2}=0$$ ...
3
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0answers
47 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
2
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1answer
25 views

Find sum of arguments where function takes supremum and infimum

Let $f(x)=(\tan x)^{\sin 2x} $ $\\$ for $x\in(0, \frac{\pi}{2})$ let $i$ be an argument where function takes infimum and $s$ - supremum. Find $i+s$ I calculated $f'(x)=2e^{\sin2x\cdot\ln{\tan ...
0
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1answer
41 views

Interchanging differentiation and evaluation

Suppose $f(x,y)\in \mathcal{C}^2$ (twice continuously differentiable, also real). When can you say $$f_x(x,y)\Biggr|_{y=0} = \partial_x\left(f(x,0)\right)$$ with the loosest possible restrictions? ...
3
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2answers
71 views

Understanding higher dimensional derivatives

I'm having trouble understanding higher dimensional derivatives. Suppose $f: \Bbb R \to \Bbb R$. We say $f$ is differentiable at $x = c$ if $\lim \limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$ ...
3
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3answers
94 views

Can the third derivative tell me anything about the original function?

The first derivative can tell me about the intervals of increase/decrease for $f(x)$. The second derivative can tell me about the concavity of $f(x)$. So can the third derivatives, and any ...
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2answers
25 views

What does $\frac{3x^2 + 2ax + b}{2y} \quad$ mean w.r.t the derivative of $y^2 = x^3 + ax^2 + bx$?

We know that if the given equation were $\quad y = x^3 + ax^2 + bx$, $\quad$ then the derivative would be $3x^2 + 2ax + b$. Since the given equation is different so the derivative will be: $$2(x^3 + ...
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2answers
57 views

What is the derivative of a matrix w.r.t itself?

what is the derivative of \begin{equation}\partial \frac{x^TVx}{\partial V} \end{equation} where V is a matrix and x is a vector. In general what is the right way to calculate matrix derivatives w.r.t ...
3
votes
1answer
86 views

Proving the Implicit function theorem in a particular case.

Let $g:\Bbb{R^2}\rightarrow\Bbb{R}$ differentiable. Assume that $g(0,0)=0$ and $g'_y$ continuous at $(0,0)$ and $g'_y(0,0)>0$. It is asking to prove the Implicit function theorem in this ...
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2answers
31 views

The equation of the normal to the curve.

The equation of the normal to the curve: $f(x)=x^2-5$ at $x=2$ I know that $y-y_1=m(x-x_1)$ But I don't really know how to procede
2
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1answer
59 views

How does this “integration by differentiation” method work

Apparently, the integral of a function f(x) from a to b can be done through differentiation through this method: $$ \int_a^b f(x)dx = \lim_{x \rightarrow \ 0 } f(\frac{d}{d x} ...
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2answers
57 views

Proof that the derivative of a function $f$ and $g$ are equivalent $\forall x \in$ the domain of $f(x)$ and $g(x)$

Set $ g(x) = \left\{ \begin{array}{lr} \frac{1}{x} & : x > 0 \\ \frac{1}{x} + 1 & : x < 0 ...
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0answers
32 views

Question about derivative notation

So i am studying for my calc test and i have a quick question does $dy/dx$ means $y'(x)$? and does $dy/dt$ means $y'(t)$? Thanks
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1answer
42 views

what is the name of the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, for every $n \geq 0$

I am solving for $x^x = 100$, I found a solution that used $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ for every $n \geq 0$ as the working equation in finding the value of $x$. I want to know what ...
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4answers
47 views

Derivative of $ \frac {\sqrt{x^2+1}}{2}$

So the wolfram alpha says the answer is: $ \frac {x}{2 \sqrt{x^2+1}}$ But when I try to take the derivative of: $ \frac {\sqrt{x^2+1}}{2}$ by quotient rule, I get: $ \frac {\frac{1}{2} \cdot ...
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1answer
23 views

Can't solve second order ODE with variation of parameters or undetermined coefficients

I have to solve $$y''+4y' +y=\frac{e^\left(-2x\right)}{x^2}$$ The homogenous equation is easy enough to solve and I got $$y(x) = c_1e^{-2 + \sqrt{3}} + c_2e^{-2 - \sqrt{3}}$$ Doing variation of ...
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0answers
39 views

Derivative Notation as a factor

In the figure below, the author uses the notation D to replace the Leibniz's notation d/dt, and after that he rewrites the equation, disconnecting the symbol of the derivative, D, of its function, ...
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1answer
26 views

There are two points on the curve given by the equation$ x^2+y^2-xy+5y+2x = 8$

There are two points on the curve given by the equation $x^2+y^2-xy+5y+2x = 8$ at which the tangent to the curve is at an angle of $\pi/4$ to the x-axis. Find the equation of the straight line joining ...
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0answers
65 views

Does $\lim_{x\rightarrow\infty}[ f(x)+f'(x)]=0$ imply $\lim_{x\rightarrow\infty}f(x)=0$? [duplicate]

Let $f$ be a continuous function with continuous derivative such that the $\lim_{x \to \infty}[f(x)+f'(x)]=0$. Is it true that the $\lim_{x \to \infty}f(x) = 0$? Thanks for your help.
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0answers
35 views

Solving a differential equation for all values of $x$

I am trying to solve the equation $ y^{\prime\prime}+\frac{{y^{\prime}}^{2}}{y}+\frac{2y^{\prime}}{x+1}+\frac{y^{\prime}}{x(x+1)}=\frac{x+1}{6xy^{2}} $ I was successful to solve the above equation ...
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1answer
40 views

How to calculate this derivative.

How to get the last equation from the above set of equations. In the book, it is mentioned straight away that the value of $P_0(t)$ is equal to that. I don't understand this step. How is it ...
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0answers
39 views

Fractional derivative definition

Suppose that $f(x) \in C^1$ for a $x \in [a, x]$. Then a regularization of Riemann-Louisville fractional derivative is defined as: $ \frac{1}{\Gamma(1-b)} \frac{d}{dx} \int_{a}^{x}\left( ...
1
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1answer
33 views

Computing differentiation rule with error bound

I have values for $x$, $f(x)$ and fixed error bounds for $f^{(n)}(x)=c_n$ for $n=\{1,2,3,4,5\}$. I want to compute $f^\prime(x)$ using $f(x-h),f(x)$, and $f(x+h)$. Since the function is continuous in ...
3
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0answers
43 views

Maximum value of an integral.

Define $$f(x)=\int_0^1e^{|t-x|}dt$$ I have to find the maximum value of $f(x)$ when $0 \leq x \leq 1$. To remove the modulus, I wrote $$f(x)=\int_0^xe^{x-t}dt + \int_x^1e^{t-x}dt$$ ...
0
votes
2answers
91 views

How do we determine if $f '(0)$ exists [duplicate]

Suppose that f: $\mathbb{R} \to \mathbb{R}$ is continuous and $f '(x)$ exists $\forall x \gt 0$ and $\lim_{x\to 0} f '(x) = 3$. Does $f '(0)$ exist? So it's apparent that my function $f$ is ...
1
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2answers
51 views

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$. The Mean Value Theorem states: a function $f$ which is continuous on the closed ...
0
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3answers
43 views

Differentiability of $f(x)=sin(x)/x$ if $x\ne0$ and $1$ if $x=0$

I am trying to see if $$f(x)= \begin{cases} \frac{\sin(x)}x &\text{ if x}\neq0\\ 1 &\text{ if x}=0. \end{cases} $$ is differentiable more than once. This is what I did: $$f'(0)= \begin{cases} ...