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

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87 views

Integral/Derivation of Modulo/Greatest Integer

I've been trying to find the integral and derivatives of the modulo function. For those of you who are not aware, it is a primarily Computer Science based operator that is defined as the remainder ...
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2answers
85 views

Why is it legitimate to perform multiplication with differentials dx?

Why is it legitimate to perform multiplication with differentials $dx$? For instance, from the statement $dy = 5dx$ one derives $\frac{dy}{dx} = 5$. I learned $\frac{dy}{dx}$ as a notation to mean ...
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1answer
31 views

Show that $\exp(x)-1=\mathcal{O}(x)$ for $x\to 0$

Find a function $g(x)$ that is as simple as possible s.t. $\exp(x)-1=\mathcal{O}(g(x))$ for $x\to 0$. Claim. Such a possible function is $g(x)=x$. Proof. Using the definition of the class $\mathcal{...
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2answers
33 views

Differential Equation, linear or non-linear?

I am new to the area of solving differential equations, and I came across the following differential equation and was wondering whether it was linear or non-linear: $dy/dx= x^3 + y^3$ I would have ...
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1answer
87 views

Minimizing $f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}}$ subject to the constraint

Let $f(r)$ be a function defined as follows \begin{align} f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}} \end{align} where $0 < A,c$ and $ t\in (0,...
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1answer
41 views

About total derivative

We all know that if $z=f(x,y)$, then the total derivative of $z$ is given by the formula $\Bbb d z = \dfrac {\partial f} {\partial x} \Bbb d x + \dfrac {\partial f} {\partial y} \Bbb d y$. My ...
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3answers
53 views

$xf'(x) = αf(x)$. How to prove that $f(x) = cx^\alpha$?

Let $f$ be a differentiable function such that $xf'(x) = \alpha f(x)$ for all $x > 0$. How do I show that $f(x) = cx^\alpha$ for some constant $c$? I have $f'(x) = \alpha f(x)/x$ , and I can see ...
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1answer
42 views

what is wrong with my derivative?!!!

I have a question which want to derivative $\left(\frac{3x-2}{5x}\right)$ what I get is $\left(\frac{3\left(5x\right)-5\left(3x-2\right)}{\left(5x\right)^2}\right)$ then $\left(\frac{15x-15x+10}{\left(...
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1answer
40 views

Word problem based on differentiation - a right circular cone.

The height of a right circular cone increases by $k\%$, its semi vertical angle remaining constant. Assuming $k$ to be small what is the approximate percentage increase in (i) total surface area (ii) ...
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1answer
18 views

Calculating differential of inverse function.

trying to find $(f'^{-1})(a)$ and am getting the wrong answer.
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1answer
96 views

Derivative definition vs its requirements for existence

In the case of an extremely disconnected function such as ${\left(-2\right)}^{x}$. One definition requires that a derivative must be continous. $(-2)^x$ has two paths that could make it discontinuous ...
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1answer
77 views

Derivative of negative log gaussian

I have a likelihood distribution $P(\mathbf{y}_i|\mathbf{W}) = \mathcal{N}(\mu, \mathbf{W}\mathbf{W}^T + \sigma^2\mathbf{I})$ I want to maximize the total likelihood for all $\mathbf{y}_i$ by ...
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1answer
353 views

If the derivative of $f$ is never zero, then $f$ is one-to-one

This is an exercise from Abbott's second edition of Understanding Analysis. Let $f$ be differentiable on an interval $A$. Show that if $f'(x) \neq 0$ on $A$, show that $f$ is one-to-one on $A$. ...
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2answers
153 views

Fréchet derivative of norm function

How to calculate the Fréchet derivative of $f: \mathbb{R}^n \to \mathbb{R}: (x_1, \dots, x_n) \mapsto \sqrt{x_{1}^{2}+ \cdots + x_{n}^{2}}$? I dont't know hot to fin a lineat operator such that $$\...
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0answers
55 views

Derivative of log trace of matrix expression

I want to find the partial derivative of \begin{equation} \log \left(X^TP(\theta)^{-1}X\right) - \log\left((Y- H(\theta)X)^T \Sigma^{-1} (Y-H(\theta)X)\right) \end{equation} with respect to the ...
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4answers
86 views

How to apply the definition of a derivative with a piecewise function?

Given the function: $$f(x) = \begin{cases} x^2+1 & \text{if $x\ge0$} \\ x^2-1 & \text{if $x < 0$} \end{cases}$$ Question: are we justified to say that the derivative at $f(0)$ exists? If ...
3
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2answers
81 views

Derivative with respect to entries of a matrix

What is the derivative of this matrix expression with respect to $\theta_k$ \begin{equation} \begin{aligned} \mathcal{J}(X, \theta) &= {\bf trace}\left( XX^TP(\theta)^{-1} \right) +{\bf trace}\...
3
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1answer
39 views

Power series differentiability at endpoints

I have the following problem: Find domain $I$ of the function defined by $f(x)=\sum\limits_{n=1}^{\infty}(3^{\frac{1}{n^2}}-1)x^n.$ Investigate differentiability of $f(x)$ in the interior of $...
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1answer
85 views

How to Solve a differential equation with both x and y?

Solve $\dfrac{dy}{dx}=\dfrac{y-3}{y^2+x^2}$ given that it passes through $(0,1)$. Right now I do not yet know how to solve differential equations with both $x$ and $y$ that you cannot separate. ...
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1answer
56 views

Function approaches zero but derivative doesn't [duplicate]

If: $y=f(x)$ and $y=0$ when $x\rightarrow\infty$ Is it possible that: $\frac{d}{dx}(y)$ is not equal to zero when $x\rightarrow\infty$ And prove it!
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1answer
35 views

Proving a Function has $f'(0) >0$ but for every $h >0$, $f(x)$ is not strictly increasing on $[-h,h]$.

From the title, consider the function $f(x) = x + x^2 S(1/x^2)$ if $x\ne 0$ and $f(x) = 0$ if $x=0$. $S(x) = 1-|x-1|$ if $-1\le x \le 3$ and $S(x) = S(x+4)$ for all real numbers $x$. I can prove ...
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2answers
33 views

if fg and f are differentiable at a, must g be differentiable at a? If not, what condition is needed to imply that g be differentiable at a?

if $fg$ and $f$ are differentiable at $a$, must $g$ be differentiable at $a$? If not, what condition is needed to imply that $g$ be differentiable at $a$? I realized that people asked similar ...
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1answer
40 views

Finding the most general antiderivative or the indefinite integrals?

A question in my Calculus book states, "Find the most general antiderivative or the indefinite integrals of the following": $$ \int \left( \frac{1}{2\sqrt x}-\frac{3}{x^4}+{4x} \right)dx $$ Can ...
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3answers
237 views

Finding points of a function's graph that are closest to a given point

A question from my calculus book states, Which points on the graph $y=4-x^2$ are the closest to the point (0,2)? Using some of my notes, I have a formula as follows (not sure what it's ...
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0answers
49 views

Calculating percentage error using differentials

I'm trying to solve the following question: The period of the pendulum of a grandfather clock is $T = 2\pi \sqrt{\frac{L}{g}}$ , where $L$ is the length (in meters) of the pendulum, $T$ is the ...
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1answer
76 views

Differentiating a contour integral

Let $P(z, t)$ be a cubic with the parameter $t$, and consider $$\mathcal{I} = \int_{\gamma(t)} \frac{dz}{\sqrt{P(z, t)}}.$$ Here, $\gamma(t)$ is a contour in the complex plane that encloses any two of ...
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2answers
23 views

Using Newton's Method to estimate a zero between a specific set of values?

One of the problems in my Calculus books states, "Use Newton's method to estimate the zero between $x=1$ and $x=2$ for the function $f(x)=X^3+2x-4$. Find the root to four decimal places." Can ...
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0answers
28 views

A tricky question using the Second Derivative Test to show that a point is a local minimizer,

Suppose we are given a function $f(x)$ with two continuous derivatives which satisfies the differential equation $xf′′(x)+3x(f′(x))^2=1−e^{−x}$ for all real x. (Do not attempt to solve this ...
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1answer
22 views

Finding all numbers [c] guaranteed by the Mean Value Theorem?

Can someone help explain how to use the Mean Value Theorem to me? I've been Googling for the last 30 minutes and still not getting anywhere. My example problem is as follows: Find all the numbers c ...
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0answers
31 views

Is there a simple solution for $ \frac{d}{dy} \log\big( \int_{0}^{y} \exp(-a \frac{x^{3/2}}{y}+ b x)\, dx \big) $

What is the solution of $$ \frac{d}{dy} \log\left( \int_0^y \exp\left(-a \frac{x^{3/2}}{y}+ b x\right)\, dx \right) $$ Where a and b are constants. Even if the solution is an approximate solution. ...
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1answer
31 views

Solving a differential equation that includes cosine

Anyone interested in coming up with a concise equation for $u(\tau)$ given the equation for its derivative below? \begin{align} \frac{du}{d\tau}=-\sigma u + S\bigg(1+B\cos(\tau)\bigg) \end{align}
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1answer
35 views

Using the 1st/2nd Derivative Test to determine intervals on which the function increases, decreases, and concaves up/down?

I have a multiple part problem in my Calculus book I'm trying to figure out while I practice for a test. The given function is $F(x)= -2x^3 + 6x^2 - 3x$, and I have the following parts solved: a. ...
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1answer
28 views

Differential of definite integral

What is the solution of $$ \frac{d}{dy} \int_{0}^{y} \exp(-y(x+1))\, dx $$ If there were no $y$ inside the exponential function the answer would be the exponential function.
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3answers
117 views

To find the nth Derivative of $9\sqrt{x}$

Can you help me to proof that the nth derivative of $9\sqrt{x}$ is $$ (-1)^{(n-1)} \cdot \frac{9(2n-2)!}{(n-1)!} \cdot (4x)^{\frac{1-2n}{2}}$$ I've tried induction but didn't go very far. Many ...
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1answer
36 views

derivative proof of $\frac{dx^2}{d x}$ is $2x$ [duplicate]

I am studying for a midterm and I have no idea about how to prove that the derivate of $\frac{d}{dx}x^2$ is equal to $2x$ Anyone has any idea? Thank you
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0answers
32 views

How can I compute the derivative

I have a smooth and bounded domain $\Omega$ an a function $$J: \mathbb{R}^n \rightarrow \mathbb{R}$$ which is non negativ, continous, has compact support in $B(0,1)$ and $$\int_{\mathbb{R}^n}J(z) d z =...
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4answers
104 views

Why isn't the derivative of $|2x^2-3x|$ equal to $|4x-3|$?

I don't quite understand why this is the case? Since when differentiating $|2x^2-3x|$ you get $\frac{(2x^2-3x)(4x-3)}{|2x^2-3x|}$...... when it is $2x^2-3x$, the derivative is $4x-3$ and when it is $-...
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2answers
34 views

Show there exists $\xi \in [a,b]$ such that $g(\xi)\int_a^\xi f(x)\text{d}x=f(\xi)\int_\xi^b g(x)\text{d}x$

Assume $f(x),g(x)$ is continuous on $[a,b]$. show that there exists $\xi \in [a,b]$, such that $$g(\xi)\int_a^\xi f(x)\text{d}x=f(\xi)\int_\xi^b g(x)\text{d}x$$ I tried to use intermediate value ...
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1answer
18 views

If $r=\sqrt{x^2+y^2}$ and $u=f(r)$ then prove $ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =f''(r)+\frac{1}{r}f'(r)$

We have $r=\sqrt{x^2+y^2}$ and $u=f(r)$ then prove , from chain rule I know that $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}.\frac{\partial r}{\partial x}$$ and $$\frac{\partial u}{...
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1answer
32 views

Ways to differentiate $(x)(x+y)$

I checked the differentiation of $(x)(x+y)$ using an online derivative tool which gives the result: $\frac{d}{dx}\left(\left(x\right)\left(x+y\right)\right) = x+y+x\left(\frac{d}{dx}\left(y\right)+1\...
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0answers
47 views

Upper and lower derivatives at local minimum

I've come across an exercise in Royden & Fitzpatrick that's got me a bit confused. It claims that if $c$ is the local minimizer for $f$ in $(a,b)$, then $$\underline{D} f(c) \leq 0 \leq \...
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4answers
101 views

Why Does $f(x) = x\sqrt{x+3}$ Only Have One Critical Point?

I am trying to find the critical points of the function $f(x) = x\sqrt{x+3}$, then by using the First Derivative Test, determine which ones are a local maximum, local minimum, or neither. Using the ...
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4answers
143 views

Find the maximum area of a right triangle with a constant perimeter P.

I have been learning calculus from a tutor and I have been trying to solve a problem that he gave me. The problem is to find the maximum area of a right triangle with a constant perimeter $P$. To ...
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0answers
15 views

Related Rates Involving a Graphical Setting

Please help me with this related rates problem: A point P is moving along the curve whose equation is $y=\sqrt{x}$. Suppose that $x$ is increasing at a rate of $4 \frac{units}{s}$ when $x=3$. How fast ...
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1answer
23 views

Examining second derivatives.

If I have a set of continuous functions X on an interval [a,b], such that f(a)=f(b)=0 for all functions in the set. Is it possible to create a mapping from X to C[a,b] by using the second derivative ...
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3answers
252 views

Two non-differentiable functions whose product is differentiable.

So I was wondering while studying analysis if there is any case where two functions aren't differential at $0$ (kind of like $1/x$) but is differentiable at 0 when combined (i.e. $fg$). I mean this ...
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3answers
34 views

Implicit derivatives and logarithmic derivatives: $\left(x^{\sqrt{x}}\right)'=?$

How would I find the derivative with respect to $x$ of $$y = \left(x^{\sqrt{x}}\right)'.$$ I can find the correct answer using the method of logarithmic differentiation that my book mysteriously ...
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1answer
33 views

Total Derivative as a Linear Map

I am currently studying Calculus on Manifolds .I am studying Spivacks book along with Munkres and J.Shurmans notes since i might not understand something from one book to another.What i noticed is so ...
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1answer
21 views

Differentiating a vector?

I am having doubt with i think a pretty simple question. I want to know how the vector in the following question id differentiated. Question: If s is a vector function, of the scalar t, whose ...
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1answer
17 views

Proving that the tangent line to the graph of $f$ at $(a, 1/a^2)$ intersects $f$ at one other point

This comes from Spivak's Calculus (problem 2b from chapter 9). Here, $f(x)=1/x^2$. My approach was as follows: I took $f'$ and wrote down the tangent line to $f$ at $(a, 1/a^2)$, namely $y_a=-2/a^...