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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1answer
29 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 ...
1
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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?
1
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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 ...
7
votes
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 ...
0
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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 $F$ ...
0
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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 $$\frac{\partial^{2015} f}{\partial x^{2015}}\text{.}$$
0
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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 ...
0
votes
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 $\frac{dy}{dx}=\frac{5x^4}{a-b}-\frac{2x}{a-b}-1$.
0
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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 ...
1
vote
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 $f$ ...
0
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
0
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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 + ...
0
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2answers
56 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
votes
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} ...
1
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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 ...
1
vote
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
0
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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 ...
-1
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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 ...
0
votes
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 ...
0
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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, ...
0
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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 ...
-2
votes
0answers
64 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.
1
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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 ...
-2
votes
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 ...
0
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0answers
38 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
vote
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
votes
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
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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} ...
1
vote
2answers
65 views

Optimization, find the dimensions of the poster with the smallest area

The top and bottom margins of a poster are 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 380 square centimeters, find the dimensions of the ...
1
vote
1answer
41 views

Application of Differentiation (Doesn't understand)

It's given the cubic equation $x^3-12x-5=0$. Show graphically that the iteration $x_{n+1}=\sqrt[3]{12x_n+5}$ should be used to find the most negative root and the positive root, and the iteration ...
4
votes
2answers
100 views

Find an upper bound for $f(x) = \sin(\sin(x))$.

I've run into this hard calculus problem that I can't seem to solve. The question is: If $f(x) = \sin(\sin x)$, use a graph to find an upper bound for $|f^{(iv)}(x)|$. I am not sure what I have ...
0
votes
1answer
32 views

Finding intervals using local min and max (in interval notation form)

I am having some trouble with the following question: Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum ...
0
votes
2answers
54 views

Solve the differential equation $z'(x) = z^2(x)$

Solve the differential equation $$z'(x) = z^2(x),$$ i.e. find $z$. Initial condition $z(0) = -1$, and the solution is in fact $$z(x) = -\frac 1{1+x},\;x>-1$$ but I don't get how they find this.
1
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0answers
24 views

Deduce the derivative from known the derivative of it composite with other.

Let $ f: \mathbb{R} \rightarrow \mathbb{R^2} $ and $ g: \mathbb{R^2} \rightarrow \mathbb{R} $ defined by $$f(t) = (t,t^2)$$ and $$g(x,y)=xy+y^2$$ Fist find $f'(t)$ and $(g \circ f)'(t)$, then ...
0
votes
1answer
60 views

How to visualize the limit of this function?

$$ f(x) = \begin{cases} x, & \text{$x$ rational} \\ -x, & \text{$x$ irrational} \end{cases} $$ $ \text{This function is not continuous at any point except 0.} $ Intuitively, I am able to ...
1
vote
2answers
27 views

Find all points on a surface which have a tangent plane parallel to given plane - is my method correct?

The question given is to find all points on the surface given by $x^3 - y^3 - 2xy - z = 0$ which have a tangent plane which is parallel to $6x - 6y - z = 0$. So, I found the two gradient vectors ...
2
votes
2answers
43 views

Find the Critical Points: $f(x) =(x^2-1)^3$

This question probably has more to do with my Algebra skills than Calculus. Nonetheless, can someone explain why the factored "term" is not set to zero (0) [second picture]. Thanks in advance.
2
votes
0answers
30 views

Can I denote $f'(g(x))=\frac{\text{d}}{\text{d}(g(x))} f(g(x))$?

Can I denote $f'(g(x))=\frac{\text{d}}{\text{d}(g(x))} f(g(x))$? Wikipedia avoids this by letting $y=f(u), u=g(x)$ and then denoting $f'(g(x))=\frac{\text{d}y}{\text{d}u}$.
1
vote
0answers
25 views

Why, using Newton's method for approximating roots, do distances have quadratic relationships?

This MIT lecture defines $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}, \forall n\in\mathbb N$, where $x_1$ is a reasonable first guess for the root of the curve in the video. It then explains how ...