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

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63 views

Maxima problem?

this question is off a textbook, and I've been having a lot of trouble with it: "A window consists of a rectangle surmounted by a semi-circle having its diameter the width of the rectangle. If the ...
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2answers
80 views

Taylor expansion

Is there an easier way to do a Taylor expansion of $e^{u^2+u}$ than do derivatives or substitute and then use Newton's binomial? For example, expanding until the $4$th term: $$e^{u^2+u}=1+u^2+u+ ...
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4answers
71 views

using definition of derivative

$f(x)=x^3-6x^2+9x-5$ is given. What is the value of $$\lim_{h\to0}\frac{[f'(1+2h)+f'(3-3h)]}{2h}$$ I tried to use the definition of derivative,and here it seems like the expression will be equal to ...
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0answers
49 views

Confusion related to the gradient of the sum of a smooth and non-smooth function

I have this confusion related to gradient. Let my function $f(x) = g(x) + h(x)$ where $g(x)$ is a differentiable function and $h(x)= \lambda\|x\|_1$ $g(x)$ is differentiable but $\|x\|_1$ is not at ...
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1answer
41 views

Derivative of logarithmic function

If the function $f(x)=\log_{2x}x^2$ is given, what is $f'(4)$? I tried to use the formula for derivative of logarithm but here the base is $2x$, so it made me confused. Note that the answer is ...
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2answers
41 views

find $\frac{\partial f(u(x(t),y(t)),v(x(t),y(t)))}{\partial t} $

how to find $$\frac{d z}{d t} $$ where is $z=f(u(x(t),y(t)),v(x(t),y(t)))$ and if some one can give me some advice which make me deal more easy with this subject
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2answers
82 views

derative of Taylor expansion

I'm reading this part of article about key points localization in image processing, and there is something I don't quite understand, mathematically it's this, $$D(w) = D + {\frac{\partial D}{\partial ...
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0answers
169 views

How to determine that a certain eigenvalue is doubly degenerate?

Given a symmetric matrix $X$. I ask myself, how to determine that a certain eigenvalue $\lambda$ is (exactly) doubly degenerate? I thought about several approaches: Calculate the derivative of $ ...
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1answer
398 views

The $n$-th derivative of the product of three functions

I would like to ask what are the general formula for the the $n$-th derivative of the product of three functions? For the product of two functions, we have the Leibniz's rule for the general formula, ...
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0answers
22 views

What is the sum of subgradients of two functions

Lets say I have two functions, one is differentiable lets say $x^2$. Another is non differentiable |x|. What is the sum of the gradient of the two functions wrt x. For $x^2$ it is 2x. For |x| it ...
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1answer
106 views

$L^p$ derivative vs normal derivative.

Let $f, g : \mathbb{R} \rightarrow \mathbb{C}$ be Lebesgue measurable functions, and let $1 \leq p < \infty$. If $f, g \in L^p$ and $$ \large \lim_{y \rightarrow 0} \normalsize \left\| ...
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1answer
241 views

Complex differentiation under the integral sign (Ahlfors)

In Ahlfors' Complex Analysis text, page 202, he claims that in $\{ \Re z>0 \} $ $$\frac{d}{dz} \int_0^\infty \frac{2 \eta}{\eta^2+z^2} \frac{\mathrm d \eta}{e^{2 \pi \eta}-1}=- \int_0^\infty ...
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1answer
205 views

Calculus, stationary point and sketching the curve

i just need help with a tiny question. Im doing caculus and is told to find stationary point and sketch curve. I have determined that there are no stationary points. How do i graph this? The original ...
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0answers
30 views

Clarification in deriving a gradient of a particular cost function.

I've been working through reading a paper by Laurens van der Maaten (http://siplab.tudelft.nl/sites/default/files/vandermaaten08a.pdf) At the end of the paper, he derives his gradient to the cost ...
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1answer
34 views

A question on functional equations.

Question: If it is given that $$ e^xf(x) = 2 + \int_0^x\sqrt{1+x^4}\,dx $$ then what is the value of $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $ Where I am stuck: Now, since we are to ...
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1answer
26 views

Several function conditions

It was told to me an exercise: Represent a continuous and differentiable function whose derivative is annulled in points $A(-1, 4)$ and $B(2, -3)$ and that complies these conditions: ...
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2answers
151 views

Why is differential calculus built on open sets?

For example in W. Rudin: Principles of Mathematical Analysis in every theorem or definition regarding the derivatives of a function from $\mathbb{R}^n \to \mathbb{R}^m$ there it always says at the ...
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2answers
342 views

Intersection points of two polynomials

How to prove that two distinct polynomial functions of degree m and n, respectively,the graphs intersect in at most $max(m,n)$ points.
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5answers
129 views

A derivative of an integral. Homework doubt

I need to solve this: $$\frac{d}{dx}(\int_{0}^{x}\sqrt{t^2+1}dt)$$ So, first I solve the integral for $(0,x)$, only need to evaluate in $x$: $$\frac{d}{dx}(\frac{x\sqrt{x^2}}{2}dt)$$ Then I solve ...
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2answers
113 views

Derivative of $\frac {x\cdot\left(1 - 3x\right)}{\sqrt{x-1}}$

Problem. Find the first derivative of $$ \dfrac {x \left( 1 - 3x \right)}{\sqrt{x-1}} $$ Work. Let $u = x-1$ and $y = \dfrac {(u+1)(-3u-2)}{\sqrt{u}} $ Using the chain rule, I ...
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2answers
47 views

Non integral degree derivatives

Could non integral degree derivative somehow be interpreted? What I mean: $f^{(1)}(x) = \frac{df(x)}{dx}$ $f^{(2)}(x) = \frac{d^{2}f(x)}{dx^2}$ How could $f^{(1.5)}(x)$ be interpreted?
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3answers
219 views

How to find the second derivative of $2x^3 + y^3 = 5$?

I'm finding it difficult to find the second derivative of the following equation: $2x^3 + y^3 = 5$. My answer is $\dfrac{-4xy^3 - 8x^4}{y^5}$, but the answer key says $\dfrac{-20x}{y^5}$. My friends ...
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1answer
78 views

Very basic derivative question

Why is the derivative of $e^x + e^{-x}$ equal to $e^x - e^{-x}$ ?
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175 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
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4answers
186 views

Derivatives: Interesting (unexpected?) situations where they arise?

I am re-learning Calculus. Can anyone provide any interesting (unexpected?) situations where Calculus derivatives arise in various situations or real-life careers? I am looking for something more ...
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1answer
86 views

Help with differential problem

The equatorial radius of the earth is approximately 6370 kilometers. Suppose that a wire is wrapped tightly around the earth's equator. Use differentials to determine approximately how much this wire ...
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4answers
369 views

Derivative of $\sin (2x)\cos(3x)$

What is the derivative of $\sin (2x)\cos(3x)?$ Is it $\cos (2x) \cos(3x)-\sin (2x) \sin(3x)$ or $2\cos (2x) \cos(3x)-3\sin (2x) \sin(3x)?$ The answer is given the latter but I only get the ...
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3answers
4k views

Derivation of Newton's second law of motion. [closed]

Can somebody help me determine the derivative and proof of Isaac Newton's second law of motion ($F=ma$)? How can this be done using calculus?
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1answer
69 views

Multiple variables in derivative and finding the slope

I'm working on slope and geometrical differentiation. The problem is to find the slope of any point on the curve $$\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1$$ I have found its derivative: $\frac{dy}{dx} ...
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1answer
107 views

Notation of derivatives…

I asked my teacher the difference between this notations. (1) $$\frac{dy}{dx}$$ (2) $$\frac{\delta y}{\delta x}$$ (3) $$\frac{\Delta y}{\Delta x}$$ He told me that there is no difference. I really ...
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1answer
139 views

finding value of constant such that function has distinct root

we have the function cubic function $$ x^3 -12x +k =0 $$ it has distinct root in $$ [0,2{]} $$ that task given to us is to find the the value of k satisfying the above conditions I proceeded ...
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3answers
97 views

Equivalent form of derivative as limit?

I was traditionally taught the formula for the derivative to be: $$ \dfrac{df}{dx} = \lim_{\Delta x \to 0} \dfrac{f(x + \Delta x) - f(x)}{\Delta x}$$ Is this an equally valid form? How can I see one ...
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2answers
133 views

shortest distant of curve from origin

what'd be shortest distance of curve from origin(0,0) function is $$ y=\frac{e^x+ e^{-x}}{2} $$ I tried taking some x and y points on curve then using distance formula finding distant then i found ...
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4answers
203 views

Using the definition of the derivative prove that if $f(x)=x^\frac{4}{3}$ then $f'(x)=\frac{4x^\frac{1}{3}}{3}$

So I have that $f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ and know that applying $f(x)=x^\frac{4}{3}=f\frac{(x+h)^\frac{4}{3} -x^\frac{4}{3}}{h}$ but am at a loss when trying to expand ...
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3answers
83 views

Finding derivative of given function.

f(t) = $\int_{t^2}^{4}\sqrt{\cos(x)+12}\;dx$ Rearranging limits of integration... $f(t) = -\int_{4}^{t^2}\sqrt{\cos(x)+12}\;dx$ Taking derivative... $f'(t) = -\sqrt{\cos(t^2)+12}\; - ...
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2answers
109 views

Differentiation of logarithmic functions using the chain rule

What's the derivative of $x^2(\ln(x^2))$? I'm having a really hard time with logarithmic differentiation. Can someone help rationalize it for me?
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2answers
111 views

Exam question: Multivariable calculus, differentiation

I've decided to finish my education through completing my last exam (I've been working for 5 years). The exam is in multivariable calculus and I took the classes 6 years ago so I am very rusty. Will ...
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1answer
197 views

How to restrict Lagrange multiplier on positive values?

Here's the function that i want to optimize: $$f(x,y) = x-2y$$ and the constraint is: $$g(x,y) = x^2 + y - 10 = 0$$ Solving with Lagrange multiplier I get: $$F(x,y) = x-2y - x^2\lambda - y\lambda ...
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3answers
131 views

Differentiable function, with $f'(x)=[f(x)]^2$

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(0)=0$ and $\forall x \in \mathbb{R}$, we have $f'(x)=[f(x)]^2$. Show that $f(x)=0, \forall x \in \mathbb{R}$.
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165 views

Derivative Riccati-Bessel function

I have found two derivatives of the so-called Riccati-Bessel functions in a textbook $$ (x j_n(x))'=xj_{n-1}(x)-nj_{n}(x)$$ and $$ (x h_n^{(1)}(x))'=x h_{n-1}^{(1)}(x)-n h_n^{(1)}(x)$$ so $j_n$ is ...
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4answers
218 views

Differentiate $\ln(\cos2x)$ With respect to $x$.

I need to differentiate $\,\ln(\cos2x)$. Can someone please explain how to do this question? Thank you.
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2answers
103 views

Differential of normal distribution

Let $$f(x)=\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sigma\sqrt{2\pi}}$$ (Normal distribution curve) Where $\sigma$ is constant. Is my derivative correct and can it be simplified further? ...
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1answer
116 views

Trigonometric Functions. Definite Integrals

Find, correct to one decimal place, the value of $$\int_{0}^{60} 2\sin(x/2) \, dx.$$ Can someone please show me how this question is done. It would be very helpful thanks!
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3answers
260 views

A question on generalization of the concept of derivative

I am looking for some material to understand the process of generalization of the concept of derivative. I would not like to just read and apply the definition of the concept of differentiation in ...
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1answer
17k views

Slope of line tangent to a curve at a given point, using First Principles

Find, from first principles, the gradient of the tangent to the curve $y = 5 - x^2$ at the point $(1,4)$ on the curve. So I'm currently lost on this question can some one please show me the solutions ...
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2answers
81 views

Why do we have $\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt$?

Why is the following representation true? $$\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt,$$ where $\psi^{(m)} (z)$ denotes the Polygamma function.
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1answer
68 views

Taylor Expansion with Integral Remainder Question

I have the following question at hand and I have to admit that I am not used to integral remainder form of taylor approximation. I am still trying to work around, so a couple of hints would be useful ...
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2answers
70 views

Finding x when slope = 1

I've been working out some problems relating to slope on the points of a curve. I'm having issues with this one: In the curve to which the equation is... $$x^2 + y^2 = 4$$ find the value of $x$ at ...
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2answers
80 views

Simplifying second derivative using trigonometric identities

Given that $x=1+\sin(t)$ , $y=\sin(t) -\frac{1}{2} \cos(2t)$ show that $\frac{\text{d}^2y}{\text{d}x^2}=2$. I am having trouble proving this. Here is my working so far: ...
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2answers
45 views

Derivative problem

Could anyone help me? Let $f:[0,1] \to \mathbb{R}$ be a function of class $C^1$ such that $f(0)=0$ and there exists $a \in ]0,1[$ with $f(a)f’(a)<0$. Show that there exists $b\in ]0,1[$ with ...