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

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42 views

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$? I know it is true when $S$ is open convex, or open connected, but what about any arbitrary $S$?
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2answers
25 views

Finding tangent vector for a curve at a given point

For a problem like this where I have to find the tangent vector at the point $(0,0,1)$: $$r(t) = \sin(t)i + (t^2 − t)j+\sqrt{1 + t}k$$ I know I would take the derivative of the problem, but would I ...
2
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1answer
38 views

Existence of complicated convex functions

In Stochastic Finance: An Introduction In Discrete Time (by Follmer, Schied), page 400, I found the following proposition: Proposition A.4. Let $I\subseteq\mathbb R$ be an open interval and ...
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0answers
23 views

Solving this ODE through derivatives?

I'm currently trying to solve a more complex system of ODE. In order to understand the environment more, I've simplified stuff a bit, such that I have arrived at $$ \alpha F(x) = G(x) + ...
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2answers
39 views

Compute $\lim_{x \rightarrow +\infty} \frac{[\int^x_0 e^{y^{2}} dy]^2}{\int^x_0 e^{2y^{2}}dy}$

I've tried to apply L'hopitals rule on this one, as this get's $\frac{\infty}{\infty}$ $$\lim_{x \rightarrow +\infty} \frac{[\int^x_0 e^{y^2}\mathrm{d}y]^2}{\int^x_0 e^{2y^2}\mathrm{d}y}$$ ...
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0answers
73 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
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1answer
37 views

Bound on the sup norm for derivatives of a particular $C^\infty$ function

I'm reading textbook "A Primer of Real Analytic Functions" and on page 86 the following "obvious" claim is made: Let $|| \cdot ||$ be the sup norm on $[0, 2 \pi]$ and define function $f$ to be ...
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0answers
38 views

Holomorphic functions (continuity of partial derivatives)

Let $f:\Omega\rightarrow \mathbb{C}$ be an holomorphic function i.e. for any $z_0\in \Omega$ there exists the limit: $$f^{'}(z_0) = \lim_{z\mapsto z_0}\frac{f(z)-f(z_0)}{z-z_0}.$$ Let us write $f(z) ...
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4answers
84 views

Suppose that $F'(x)\leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x)\leq G(x)$ for all $x \in \mathbb{R}$.

Suppose that $F'(x) \leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x) \leq G(x)$ for all $x \in \mathbb{R}$. Prove or disprove. I came up with counterexample that $\cos(x) \leq \cos(x)+1 $ for all ...
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0answers
22 views

Conflicting information about second derivative test and graphing

When performing a first derivative test, you have a point where the first derivative is $0$ or undefined. If the values to the left of that point are increasing and those to the right are decreasing, ...
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2answers
24 views

Optimization - Maximizing Profit

I have been struggling with the problem below for quite some time now and no one can seem to figure it out, so I am asking it here. The question is as follows: You own an apartment complex with 50 ...
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1answer
84 views

What is the derivative of $\frac{x^{n+1}}{n+1}$?

I am ask to find the most general antiderivative of $f(x)= x^n$ where $n \geq 0$. However, I wondering how the derivative of $\dfrac{x^{n+1}}{n+1}$ is equal to $x^n$ My answer is $x^n - x^{n+1} $ ...
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2answers
73 views

What is the Derivative of x^x [duplicate]

I was browsing through my old textbook and I found this problem: Find Derivative of $ x^x$ My work : Haven't got a clue yet, where to start?
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2answers
31 views

Is Lagrange's mean value theorem is valid on $x^{1/3}$ in $[-1,1]$?

Suppose I have a function $f(x)=x^{\frac{1}{3}}$ in interval $[-1,1]$ , is lagrange's mean value theorem valid here ? $f(x)$ is continuous in this interval but there's a confusion in it's derivative. ...
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1answer
26 views

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$. If the directional derivatives are continuous, does this mean $f$ is differentiable?

There is a result which states that for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ if all its partial derivatives exist and are continuous then the total derivative $Df$ exists. If I ...
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4answers
77 views

If $A+B=\pi/3$ then what will maximum value of $\tan(A).\tan(B)$?

Suppose I am given that $$A+B=\frac{\pi}{3}$$ then what will be maximum value of $$\tan(A).\tan(B)=?$$ $$\tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A).\tan(B)}=\sqrt{3}$$ then ...
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1answer
65 views

Maximise $y$ with respect to $x$ for $y=\prod_{k=1}^{\infty}(1-x^{-k})$

$$y=\prod_{k=1}^{\infty}(1-x^{-k})$$ I want to maximise this function. So far I have: $$\ln(y)=\sum_{k=1}^{\infty}\ln(1-x^{-k})$$ ...
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2answers
22 views

Families of continuous functions with non-continuous derivatives

What families of functions have the property of being continuous yet having a non-continuous derivative? And how many of these families are there? $$f(x) = \sqrt[n]{x}$$ when "n" is an odd number ...
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2answers
33 views

At 2:00pm a car's speedometer reads 30mph, and at 2:10pm it reads 35mph. Use the Mean Value Theorem to find an acceleration the car must achieve.

I'm only assuming that f(a) and f(b) are assigned to each respective velocity, but I'm not sure how the mean value theorem can be applied to distance rate and time.
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3answers
57 views

How do I find the derivative of $a^x$?

The homework question I have is find the derivative of $8^{log_2(x)}$ but we haven't learn't how to find the derivative of a function where the variable is the power of a constant. How do I do this?
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0answers
33 views

Derivative of the Gauss map is zero

If the derivative of the Gauss map is zero in every point in the image of a given local chart, can I conclude that the normal vector is constant and such image is contained in a plane? Edit: The ...
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2answers
29 views

Straight vs Partial derivative

Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?
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1answer
29 views

Finding concavity using a second derivative that is never greater than zero.

When is $\frac{3x-8}{4(2-x)^{3/2}} > 0$? The equation above is the second derivative of the function: $$f(x) = x\sqrt{2-x}$$ I am wanting to find the concavity of the original function. I know ...
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2answers
30 views

Using the binomial theorem to generate a geometric proof of the derivative.

According to wikipedia, if we wanted to prove $$(x^n)'=nx^{n-1}$$ geometrically by creating an $n$-dimensional hypercube $$(x+\Delta x)^n$$ and setting $a=x$ and $b=\Delta x$, we could expand using ...
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1answer
27 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
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3answers
53 views

Find the derivative of $2^x-3^y=1$ and then find the limit of $dy/dx$ as $x\to\infty$

Please tell the answer If $2^x-3^y=1$ then what is the value of $$\lim_{x\to\infty} \frac {dy}{dx}?$$ I have tried finding the derivative implicitly, but I only get $0$ on both sides.
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1answer
22 views

Vertical asymptotes of a given non-rational radical funtion

We have that $f$ is a function $f(x) = x\sqrt{x+4}$. Hence, $f'(x) = \dfrac{3x+8}{2\sqrt{x+4}}$. Then, $\lim_{x \to -4^+}f'(x) = -\infty$. This means that $f$ has a vertical slope at $f(-4)$. It ...
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2answers
81 views

Derivative of a function with respect to a matrix

I need to calculate derivative of the following function with respect to the matrix X: $f(X)=||diag(X^TX)||_2^2$ where $diag()$ returns diagonal elements of a matrix into a vector. How can I ...
2
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3answers
97 views

Derivative of sin(x)/x at $0$ by definition of derivative

the question I am attempting is: Show $f '(0) = 0$ for: $$f(x) = \left\{ \begin{array}{lr} \frac{\sin(x)}{x} & : x \neq 0\\ 1 & : x=0 \end{array} \right.$$ So I got stuck after the ...
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1answer
87 views

How can I prove that no solution (except y=0) cannot be extended for all t axis?

Given this ODE: $$\dot{y} = (2 + \cos y(t)) \cdot{y^2(t)}$$ I need to show that no solution of this equation holds for every $t \in \mathbb{R}$, and I don't even know how to start. Any hints will do.
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2answers
338 views

A differentiation question conceptual query

I'm quite unsure about how to deal with differentiation of absolute functions, and their continuity. For example, the question I was dealing with was the following: $$ f(x) = \frac{x}{1 + |x|}$$ ...
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1answer
19 views

Related Rates Question Concerning Boyle's Law

Boyle's Law for enclosed gases states that if the volume is kept constant, the pressure P and temperature T are related by the equation P/T=k, where k is a constant. Suppose that the rate of change of ...
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2answers
44 views

If $(g'(x))^2 = g(x)$…

The question is if $(g'(x))^2 = g(x)$ for all real $x$ and $g(0) = 0$, $g(4) = 4$, then $g(1)$ equals... The answer is 1/4. I was thinking if you plugged in the numbers, then that would mean $g'(0) ...
0
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1answer
29 views

Why $f_{xy}(x_c,y_c)=f_{yx}(x_c,y_c)$?

Why $f_{xy}(x_c,y_c)=f_{yx}(x_c,y_c)$? I found this used here (in the definition of B). $x_c,y_c$ are critical points.
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2answers
34 views

Finding Equation of tangent line

Can someone double check my work to see if I'm doing it correctly? Find the equation of the line tangent to the graph of $(2,1)$ where $f$ is given by $f(x) = 2x^3 - 2x^2 + 1$ 1) $f'(x) = 6x^2-4x$ ...
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2answers
38 views

how to find tangent line at a given point, without equation

Find the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form. I don't know how can I get the tangent line, without a ...
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1answer
17 views

Using Taylor's Theorem to expect maximum error

Suppose that we use $p(x) = 1 + x + \frac {x^2} 2$ as an approximation for $f(x) = e^x$ on the interval (-.5, .5). What's the maximum error we can expect. I know that Taylor's Theorem states Let ...
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2answers
52 views

Calculus Finding the derivative

I have a practice problem that says: Let $$j(x) = \frac{g(x)}{f(x)}$$ Find $j'(1)$: I don't know how to do this. The answer the book has is $-2$. What I tried to do was set $j(0) = ...
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1answer
48 views

Complex function of class $C^m$

Let $m$ a positive integer and consider the function $$f(z)=\vert z\vert^\alpha z$$ with $\alpha>0$. I have to find the value of $\alpha$ for which $f\in C^m(\mathbb{C},\mathbb{C})$. Now if ...
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2answers
43 views

How do I see that a linear function $L: \mathbb R^n \rightarrow \mathbb R^m$ is smooth?

How do I see that a linear function $L: \mathbb R^n \rightarrow \mathbb R^m$ is smooth ? I see that $L$ is indeed differentiable with $(DL)_x = L$ by definition of the derivative. But how do I ...
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1answer
66 views

Differential identity $\left(x^2\frac{d}{dx}\right)^nf(x)=x^{n+1}\frac{d^n}{dx^n}\left(x^{n-1}f(x)\right)$

I have found the following differential identity: $$\left(-x^2\frac{d}{dx}\right)^nf(x)=(-1)^n x^{n+1}\frac{d^n}{dx^n}\left(x^{n-1}f(x)\right)$$ I have used it to find an alternative Rodrigues ...
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3answers
74 views

Proving that the second derivative of a convex function is nonnegative

My task is as follows: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of ...
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3answers
3k views

Why, although these functions have the same derivative, do they not differ by a constant?

I calculated the derivative of $\arctan\left(\frac{1+x}{1-x}\right)$ to be $\frac{1}{1+x^2}$. This is the same as $(\arctan)'$. Why is there no $c$ that satisfies $\arctan\left(\frac{1+x}{1-x}\right) ...
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0answers
51 views

Prove there exists a infinitely differentiable function whose value of partial derivatives of all orders at $0$ is a given function

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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vote
4answers
399 views

Could such a polynomial/rational function exist? [closed]

Is it possible for there to be a function, which has "nice values" for its $x$-intercepts, local extremes, and inflection points? By nice values, I essentially mean that all the $x$-intercepts, local ...
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2answers
34 views

Parametric differentiation

The parametric equations of a curve are $$\begin{cases}x(t)=e^{-t}\cos t\\y(t)=e^{-t}\sin t\end{cases}$$ Show that $$\frac{dy}{dx}= \tan\left(t-\frac{\pi}{4}\right)$$ I did the differentiation ...
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0answers
26 views

Taylor expansion of a power function

I was wondering about Taylor expansions of functions of the form $x^p$, where p is a real number, about $x = 0$. It seems clear how to do it about any other point, but what happens to the series as I ...
2
votes
3answers
44 views

differentiation of a matrix function

In statistics, the residual sum of squares is given by the formula $$ \operatorname{RSS}(\beta) = (\mathbf{y} - \mathbf{X}\beta)^T(\mathbf{y} - \mathbf{X}\beta)$$ I know differentiation of scalar ...
2
votes
2answers
60 views

Determining the rate of change of a radius as a sphere loses volume

Problem: A spherical balloon leaks $0.2\mathrm m^3 / \mathrm{min}$. How fast does the radius of the balloon decrease the moment the radius is $0.5\mathrm m$? My progress: Since we're dealing with ...
9
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6answers
129 views

How to differentiate $y=\sqrt{\frac{1+x}{1-x}}$?

I'm trying to solve this problem but I think I'm missing something. Here's what I've done so far: $$g(x) = \frac{1+x}{1-x}$$ $$u = 1+x$$ $$u' = 1$$ $$v = 1-x$$ $$v' = -1$$ $$g'(x) = \frac{(1-x) ...