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Questions tagged [derivatives]

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

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Taking derivative with chain rule

Suppose I have a function: $f(x(\eta),\eta)$ and I want to take the derivative with respect to $\eta$. Note that $f$ is a function of $x$ and $\eta$ and that $x$ itself is a function of $\eta$. I am ...
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1answer
52 views

property of the tangent line.

If the traditional way to define the tangent line to a curve $f(x)$ through the point say $(a , f(a))$ is: ( the tangent line through the point $(a ,f(a))$ is the line that passes through this point ...
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36 views

Application of Taylor's theorem: find upper bound for remainder?

Suppose $f$ is a $C^2$ function with compact support. I.e. $f$ is $0$ outside a closed interval. Then $f,f',f''$ are uniformly continuous and bounded on $\mathbb{R}$. My textbook then claims that the ...
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1answer
22 views

How to find a partial derivative in order to check whether the function is differentiable

I need to find out whether the following function is differentiable at the point $(0,0)$. $$ f(x)=\sqrt[3]{1+|x|^{2/e}\cdot|y|^{3/\pi}} $$ I think I need to find the partial derivatives first, but the ...
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1answer
37 views

Is integral of a function differentiable?

If we have a continuous function $f(x)$ and its integral is $F(x)=\displaystyle \int_a^x f(x)\ dx$, will $F(x)$ be differentiable?
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1answer
26 views

Complex differentiable function with non-continuous partial dervative [on hold]

I'm looking for a complex-valued function $f$ which is complex differentiable in $z_0$ but where the partial-derivatives are non-continuous in $z_0$. Can someone give an example? Best! Annette
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3answers
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Is a function differentiable at a point if its derivative is continuous at that point?

My professor said that the title statement might not always be the case and gave $$x^2 \sin\left(\frac{1}{x}\right)$$ at $x=0$ as a counter-example. But I don't seem to understand its ...
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0answers
8 views

Matrix derivative in images matching problem

Problem Suppose zero-centered matrices $\mathbf{X}$ and $\mathbf{Y}$ of shape $\mathbb{R}^{n\times 2}$. Each row of $\mathbf{X}$ and $\mathbf{Y}$ represents a point on 2-D plane. Therefore, they each ...
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10 views

How to create a cubic spline between the lines x=0 and y=1?

I am trying to create a simple cubic spline from points (0,0) to (m,1) connecting the lines y=1 and x=0. However, I am having trouble getting the spline to be tangential to the x=0 line at (0,0). ...
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0answers
3k views

How to Use Differentials to Estimate the Percentage Change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$

I am trying to determine how to use differentials to estimate the percentage change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$. So far, I have done the following steps: 1) Determine ...
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Dual via KKT for convex problem

I would appreciate a bit of help in order to continue simplifying the following problem. I have a primal convex problem $ \mathcal{P} $ and I am trying to find its dual $ \mathcal{D} $. Due to the ...
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2answers
69 views

Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$ f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2} $$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
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5answers
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Is my solution of $\frac{d }{dx}\int_0^{\cos x}\sqrt{1+t^4}dt$ correct?

$$\frac{d }{dx}\int_0^{\cos x}\sqrt{1+t^4}dt$$ $$\frac{(\sqrt{1+\cos^4x}-1)dx}{dx}$$ $$\sqrt{1+\cos^4x}-1$$ The answer seems weird to me, but I see no other way to do this. Was this correct? If not, ...
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2answers
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How many critical points are there in $(\,x- 1\,)(\,x- 2\,)\,…\,(\,x- 2020\,)$?

How many critical points are there in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 2020\,)$$ My observation is There are $7$ critical points (${\rm C.P}$) in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 8\,)$$ $[\,{\...
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0answers
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Derivation,Series,partial derivative! [on hold]

In this question & is a partial derivative. D.D.D=µ[&3-(1/12+1/16)&5+….] HOW IT CAN BE SOLVE.I Am not understanding the basic step how it may proceed
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1answer
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Derivate of absolute value of complex valued function

I have a derivate where $a(z)$ is complex valued. $$\frac{da(z)}{dz}=-\Delta a(z)-\Delta^*e^{-2i\omega z/\bar{c}}b(z)$$ where $\Delta=\frac{\sigma}{2\bar{\zeta}}-\frac{i\omega\nu}{2\bar{c}}$ and star ...
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5answers
16k views

Differentiating both sides of an equation

I'm going through the MIT lecture on implicit differentiation, and the first two steps are shown below, taking the derivative of both sides: $$x^2 + y^2 = 1$$ $$\frac{d}{dx} x^2 + \frac{d}{dx} y^2 = \...
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4answers
80 views

differentiation of fractional part of $x$

What is the differentiation of fractional part of $x$? Since the slope of $\{x\}$ is $1$ so that derivative of $\{x\}$ should be $1$. Is it correct or not
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1answer
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Derivative of product and summed function

I am trying to take the derivative with respect to $x$ of the following function: $$ F(x) = \sum_{i} ax^i(1-bx^j)\prod_k(1-cx^k) $$ With $i\in [2,n]$, $j=n-i+1$ and $k=i+1$ to $n$. I am struggling ...
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3answers
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Derivative of Binary Cross Entropy - why are my signs not right?

I'm trying to derive formulas used in backpropagation for a neural network that uses a binary cross entropy loss function. When I perform the differentiation, however, my signs do not come out right: ...
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4answers
70 views

How to differentiate $ f(x) = (\csc \pi x)^{-4/5}$ [closed]

How can I differentiate $ f(x) = (\csc \pi x)^{-4/5}$? My problem is that there is a $\pi$ in the question and it's a constant so how is it supposed to be differentiated in the first place? Would ...
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2answers
34 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...
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1answer
48 views

A Lipschitz function is $C^1$?

I am wondering if a Lipschitz function $f:[a,b]\to\mathbb{R}$ is $C^1$, that is its derivative is also continuous? I have seen that in a text however I could not prove it and does not seem so obvious ...
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7answers
915 views

Using chain rule to differentiate $f(x)=a(x)b(x)$?

Why can I not apply the chain rule to a product in the following way. If we have some product: $$f(x)=a(x)b(x)$$ Consider the multiplication of b by a as another’s function so that: $$f(b(x))=ab$$ ...
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1answer
1k views

The derivative of the Electric field for a uniformly charged rod

The formula for the electric field at a point due to a charge $Q$ (just considering the magnitude) at some distance $x$ away from the point is $E=\dfrac{k_eQ}{x^2}$ where $k_e$ is a constant equal to ...
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1answer
141 views

How can we have $k$ strong enough such that $\sqrt{x^{\,2}+ 3\,x+ 1}+ x= k\in \mathbb{Q},\,x\in \mathbb{Q}\,?$ [on hold]

Prove $$\sqrt[4\,]{x^{\,4}+ 1}= \sqrt{x^{\,2}+ 3\,x+ 1}+ \sqrt{2\,x+ 10} \tag{1}$$ has no real root. By W$\mid$A $$\sqrt[4\,]{x^{\,4}+ 1}> \sqrt{x^{\,2}+ 3\,x+ 1}+ \sqrt{2\,x+ 10}\Leftrightarrow ...
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4answers
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Matrix Calculus in Least-Square method

In the proof of matrix solution of Least Square Method, I see some matrix calculus, which I have no clue. Can anyone explain to me or recommend me a good link to study this sort of matrix calculus? ...
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4answers
48 views

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$ Firstly I wanted to calculate $\int \sin (t^2) dt$ and then use $x$ and $\sqrt {x^2+1}$. But this antiderivative not exist so how can I do ...
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0answers
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This paper implies that $a \frac{\partial{b^\ast}}{\partial{q}} = b \frac{\partial{a^\ast}}{\partial{q}}$ and I don't see why.

This question is regarding a particular paper that claims a particular result that I cannot seem to follow. The paper is: Cyclic Spectroscopy of the millisecond pulsar, B1937+21 (The paper should be ...
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2answers
1k views

Show function is monotonic and if it has maximum or minimum without second derivative

Hi all I have homework show function $f(x)= -x^{2}$ is monotonic and show it has maximum and or minimum and don't use second derivative for this. Please say I do wrong or not because teacher control ...
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1answer
49 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
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1answer
30 views

How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?

How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$ here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
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1answer
42 views

Does derivative imply weak derivative?

It is known that there are functions whose weak derivative exists but (classical) derivative does not exist. I want to confirm that "any differentiable function is weakly differentiable". Help me.
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24 views

Minimum number of real repeated roots of the following function are?

Let $g(x)=f’(x)$ The given figure represents the graph of $y=g(x), a\leq x \leq b$ . Given $f(c)=0$ Find the minimum number of repeated roots. Since the function is always decreasing so it crosses ...
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1answer
894 views

How to determine whether a piecewise function has a derivative?

Could someone show me a worked example of showing whether a piecewise function is differentiable at some $x=a$? I can show that it is continuous at $a$, as the limit as $x\to a$ (from both sides) ...
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1answer
29 views

What's the meaning of a derivative of a parametric curve?

A parametric curve $C$ can be defined as follows $$ C(p) = \{x(p), y(p) \}, \; p \in [0, 1] $$ where $p$ is the parameter. We can define the unnormalised tangent to the point of the curve ...
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3answers
32 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
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$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
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1answer
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Differentiation when there are continuously many variables

Suppose there are a continuum of tasks in a unit range $[0, 1]$, and for each task $i \in [0, 1]$, a firm can choose the amount of robots, $x_i$. I am hoping to get the first-order condition (e.g., $\...
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1answer
51 views

Find a function's differential

I need to prove that the following function is differentiable and find $df$. $$ f(x,y,z)=\frac{\sqrt{x^2-y^2}}{z^2+x+y} $$ I found all partial derivatives: $$ \frac{\partial f}{\partial x}=\frac{x(y+z^...
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0answers
55 views

Convex or not convex?

I would like to find out whether $ z(x) = x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x $ is convex or not, $ c > 0 $, $ x \in \mathbb{C}^{N} $, $ P_1, P_2, P_3 \in \mathbb{C}^{M \...
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1answer
25 views

Gradient and the Hessian of quartic function

I am given a symmetric $n\times n$-matrix $A$ and a vector $v$, and now I have to compute the gradient and the Hessian of $$f(x) = (x^TAx)^2-(x^TAv)^2.$$ I guess that the gradient is $$\nabla f (x)...
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1answer
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modelling of an equation

A scientist is studying a population of mice on an island. The number of mice, N, in the population after t, months of the study of the modelled is $$N=\frac{900}{3+7e^{-0.25t}}$$ They said show that ...
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2answers
46 views

Partial Differentiation of $\frac 00$

Let: $$f(x,y)=x^2y\sin\left(\frac{y}{x}\right),\ x\neq0$$ $$f(x,y)=0, \ x=0$$ Partial differentiation is obvious for $x\neq0$, however, for $x = 0$ and the derivative over $x$, one gets: $$\lim_{h\to ...
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Determine the number of local maxima und minima of this function.

$f:\Bbb R^2 \to \Bbb R:x \to exp(x^2_1+x_2^2)-8x_1^2-4x_2^4 $ Is there any smart way to determine the number of local maxima/minima of this function? We don't neet to find the exact points.
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20 views

How to derive MRTS = 1/2 < 1 = w/r when production function equals √L+2K?

I'm finding the total cost of a production function. How do I derive the marginal rate of technical substitution (MRTS) solution as 1/2 < 1 = w/r, and why is it less than (<) as opposed to = w/r?...
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1answer
55 views

Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
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3answers
40 views

Finding a particular solution of $y''+4y'+4y=\frac{e^{-2x}}{1+x}$

I am asked to find the general solution to: $$y''+4y'+4y=\dfrac{e^{-2x}}{1+x}~~~,~~~x>0$$ I have managed to find the homogeneous solution, which is: $Ae^{-2x}$ + $Bxe^{-2x}$ I am now trying to ...
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3answers
41 views

Chain rule in derivatives

Well, I have to derive this function: $$f(x)=\sin(2x \sqrt[3]{x+1} )$$ I want to use the chain rule, and I want to use it like this; I will call: $$T=x+1$$ $$Q=2x. \sqrt[3]{t} $$ $$f=\sin (Q)$$ So ...
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3answers
49 views

Finding the derivative of $ g(x) = tan(3x) $ using the definition

I was asked to find the derivative of $tan(3x)$ using the limit definition I am bit stuck at the steps, can anyone please explain ? Thank you so much , would be a great help !