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
37 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
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2answers
28 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
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0answers
19 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
114 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
50 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
30 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
42 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
87 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
168 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
32 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
votes
2answers
95 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 ...
4
votes
1answer
89 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 ...
1
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2answers
35 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
84 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
58 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
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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
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
48 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
46 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
27 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 ...
1
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0answers
37 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
41 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
56 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
47 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 ...
5
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4answers
116 views

Maximum value of the integral $\int_0^1e^{|t-x|}dt$ for $0 \leq x \leq 1$

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
99 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
62 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
45 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
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2answers
103 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
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1answer
45 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
119 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
101 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 ...
1
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0answers
25 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
66 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
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2answers
41 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
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2answers
46 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
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0answers
31 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 $f'(g(x))=\frac{\text{d}y}{\text{d}u}$.
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0answers
29 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 ...
1
vote
1answer
31 views

Absolute Maximum and Minimum of cos function

I am having a little trouble trying to figure out the following problem: Find the absolute maximum and minimum values of the function $f(x) = x-2\cos x$ on the interval $[0, 2\pi]$. I have taken the ...
6
votes
8answers
1k views

Infinite number of Derivatives

Is there a kind of function (other than trigonometric) that you can take infinite amount of derivatives without it ever becoming 0. Algebraic functions now matter how long, or how many powers it has ...
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3answers
46 views

Derivative of $(1-x)^{-2}$

Wolfram Alpha is telling me that the answer is $$-\frac2{(x-1)^3}.$$ But I thought that by using the chain rule you multiply the front by $2$ then subtract the exponent by $1$ then multiply by the ...
2
votes
3answers
567 views

The sum of two coordinates at which the first two derivatives of $f(x) = e^{2x}(x^2 + 2x)$ are equal

I came across the problem on Khan Academy while studying differential calculus: Consider the function $f(x) = e^{2x}(x^2 + 2x)$. There are two x-coordinates at which $f'(x) = f''(x)$. What is ...
0
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1answer
27 views

Does Fermat's Theorem for Stationary Points hold for functions $f: \mathbb C \to \mathbb R$

Given a function $f: \mathbb C \to \mathbb R$ ($z = x+yi, \; x,y \in \mathbb R)$ Does this hold? $f$ has an extremum at $ z_0 = x_0 + iy_0$ $f$ is differentiable at $S$ and $z_0 \in S$ ...
0
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0answers
17 views

Find function $f(x,y)$ such that $f(x,0) = J(x)$ and $\nabla_{(x,y)} f(x,y) = g'(y)h(J,\nabla_x J)$?

Let $J:\Omega \to \mathbb{R}$ be a smooth function such that $0 < C_1 \leq J(x) \leq C_2 < \infty$. Is it possible to find a function $f:\Omega \times [0,\infty)$ such that $$f(x,0) = J(x)$$ ...
2
votes
1answer
45 views

Abel's theorem for the derivative of a power series

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function, $(a_0, a_1, \dots)$ is a sequence of real numbers and $x_*$ is a positive real number, such that the following two conditions hold: for ...
0
votes
0answers
27 views

proving that a function has a third derivative

Let $f: \mathbb R \rightarrow \mathbb R$ be differentiable and non-decreasing define $h(x)= e^{f'''(x)}$ i need to show that $h$ is differentiable and non-decreasing on $\mathbb R$. i know that i can ...
1
vote
3answers
63 views

How do I know that $\ln(x^2+1)-x \arctan(x)$ is always negative or zero?

I did google the function and I can clearly see that it is always negative or zero, but I have no idea how I would have found this on my own. Both the logarithm and the $x\cdot \arctan(x)$ are ...
2
votes
0answers
108 views

About the differential notation in measure theory

Is there any good reason for which integrating according to a measure includes a $\mathrm d$ as in $\int f\mathrm d\mu$ ? Or is it just a manner to keep formal consistency with the traditional ...
2
votes
1answer
27 views

How exactly is this happening?

I was studying Derivative and my book says if: Then its derivative is: I can't understand how the writer has changed the first derivative fraction into the second one. In other words, how did he ...