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

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2answers
14 views

Evaluating a statement without calculating the indefinite integral

I'm cramming for a supplementary exam so you might see a ton of questions like these in the 48+ hours to come <3 The question is more of just a yes or no ; Evaluate the statement without ...
1
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1answer
11 views

Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: \begin{equation} Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt \end{equation} I am trying to calculate the derivative of $Γ$ with respect ...
3
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1answer
43 views

Study of differentiablity of function

Study the differentiability of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ $f(x,y)=\begin{cases} \frac{x^3+y^3}{x^2+\left|y\right|} & (x,y)\ne(0,0) \\ 0 &(x,y)=(0,0) \\ ...
7
votes
2answers
176 views

Derivative of the magnitude of a vector. Does it exist, or not?

I have a puzzling situation involving derivatives. I want to derivate: $$ \frac{d}{dx}| \mathbf F(x)| $$ This was actually something involving physics. Lets be 2-dimensional for simplicity. Let a ...
1
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3answers
31 views

Derivatives with different rules

I'm having trouble with this one problem that just deals with deriving. I can't seem to figure out how they got their answer. Any help would be appreciated! Thanks! $ \frac{(x+1)^2}{(x^2+1)^3} $ The ...
7
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1answer
112 views
+50

Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...
1
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1answer
27 views

Find the Derivatives of $g(x) = \sqrt{3-2x^2}$ and $h(x) = \ln {(x^2 – x)}$

I am asked to find the derivatives of $g(x) =\sqrt{3-2x^2}$ and $h(x) = \ln{(x^2 – x)} $ For: $g(x)h(x)$ and $\dfrac{h(x)}{g(x)}$ and $h^3 (x)$ First off I am not sure if my derivatives are ...
0
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1answer
55 views

Can this $dx$ be taken out?

I have this expression: $$\frac{1}{f(x)}\frac{df(x)}{dx}=\frac{dg(x)}{dx}$$ Can the two $dx$ be "simplified"? Namely, to get $$\frac{df(x)}{f(x)}=dg(x)$$ Is this right? If it is, what is the ...
2
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2answers
407 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to \mathcal{L}^2(X,Z)$...
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2answers
41 views

Find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$

I need to find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$ I am trying to first get y in terms of x, but that is quite lengthy and feels like I am doing something wrong. How do I go about this ...
3
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0answers
46 views

More convenient form of derivative of $\mathrm{sinc}(x)$

$\mathrm{sinc}(x)$ is defined as $\frac{\sin(x)}{x}$ except continuous at $x=0$ (insert the removable singularity). The derivative of $\mathrm{sinc}(x)$ is usually given as the derivative of $\frac{\...
0
votes
0answers
32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
0
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2answers
93 views

Derivative of $\log x$

If for the function $f$ there exists $g$ such that $g(f(x)) = x $, find the derivative of $g$. Then use this for computing the derivative of $f(x) = \log(x)$ I tried this: $(g(f(x)))´ = g´(f(x))f´(...
1
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1answer
25 views

How to calculate $∇(r^2/(2z(1+a/z^2)))$ in cylindrical coordinates

How to calculate $$∇\bigg(\frac{(ρ^2)}{2z(1+\frac{a}{z^2})}\bigg)$$ where the function is in cylindrical coordinates $$ρ^2=x^2+z^2$$ $$∇\bigg(\frac{x^2+z^2}{2z(1+\frac{a}{z^2})}\bigg)$$ Is the ...
1
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2answers
18 views

Sign of the derivative $ -e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $

Good morning to everyone. I have a problem with finding the sign of a derivative: $$ \frac{d}{dx}f(x)=-e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $$ ...
0
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6answers
32 views

Critical points of a cubic function

There is a function $x^3 - 6x^2 + 9x + 1$. Its critical points are $1$ and $3$. I am very confused, if these points are maximum and minimum points respectively or are both inflection points. Can ...
2
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0answers
86 views
+50

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
0
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1answer
15 views

Derivative of dot product with transposed function

According to this post Derivative of dot product I have a similar task: $$\langle f(x),g(x) \rangle = f(x)g(x)^T=j(x)$$ I have to show: $j'(x)=g'(x)f(x)^T+g(x)^Tf'(x)$ I know how to ...
0
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1answer
23 views

How can I prove that $(0,0)$ saddle-type inflection point of $x|y|+y|x|$?

How can I prove that (0,0) is saddle-type inflection point? the function is: $f(x,y)=x|y|+y|x|$ How can I find the second derivative by $x$ (for the hessian matrix)
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0answers
9 views

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=0.$$ ...
0
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1answer
72 views

Why can this differential equation be written in $3$ different ways?

Suppose we have the following differential equation using operator notation: $$(D-x)(D+x)y=0\tag{1}$$ where $$D=\frac{d}{dx}$$ Now I could rewrite $(1)$ as $$\begin{align}\require{enclose}(D-x)(D+x)y&...
0
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0answers
15 views

Translating Logistic Regression loss function to Softmax

I currently have a program which takes a feature vector and classification, and applies it to a known weight vector to generate a loss gradient using Logistic Regression. This is that code: ...
0
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1answer
51 views

Examples of (not) uniformly continuous, non-differentiable, non-periodic functions

Let $I\subseteq\mathbb{R}$ and $f:I\to\mathbb{R}.$ $(0)$ If $f$ is discontinuous on $I$, then it is not uniformly continuous. $(1)$ Suppose $I$ is open and bounded. If $f$ is unbounded on $I$,...
0
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1answer
30 views

Differentiation of a floor function using Mean Value Theorem

I came acroos the following function today which gave different results for its differentiation. Here is it $$f(x)=\lfloor x\rfloor\text{ at }x \in [1,2]$$. In the above function, the value of f(x) ...
3
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1answer
50 views

function is not differentiable on $\mathbb R\setminus\{0\}$

I need to prove that the given function $f$ is not differentiable on $\mathbb R \setminus\{0\}$. $$ f(x) = \begin{cases} x^2, \ x \in \mathbb{Q}\\ 0, \ x \in \mathbb{R}-\mathbb{Q} \end{cases} $$ ...
0
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0answers
35 views

Derivatives that are tangent to the original function

I was recently studying parabolas $ f(x) = ax^2 + bx + c $ whose derivative $f'(x) = 2ax + b$ is tangent to itself -- one example would be $f(x) = x^2 -6x +10;$ it is easy to see that if $c = a + \...
0
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2answers
40 views

Intuition behind the derivative of are of a square? How to properly use the derivative ?

If I derive the formula $$S=16t^2$$, where S denotes the distance and t denotes time I get $$ds/dt= 32t$$. This in return give me a formula for the speed of the object at any time t. However if we ...
1
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1answer
43 views

Derivative of $\arcsin \frac{x-1}{x+1}$

I was looking at a question that asks for the derivative of $\arcsin (\frac {x+1}{x-1}) $. The solution starts by saying $y = \frac{x+1}{x-1}$, so $1-y^2= \frac{4x}{(x+1)^2}$ and $\frac{1}{\sqrt{1-y^...
0
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1answer
46 views

How many solutions does $a^x=2016x$ for $a > 0$ have?

How many solutions does $a^x=2016x$ for $a > 0$ and $x \in \mathbb{R}$ have? Note that for $x < 0$ we have $a^x > 0$ and $2016x < 0$ so we can consider only $x \ge 0$. Let $f(x) = 2016x - ...
1
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1answer
22 views

Derivative of dot product of Residual Sum Square in matrix notation

I am trying to derive the following expression w.r.t. $\beta$: \begin{equation} RSS(\beta) = (\mathbf{y} - \mathbf{X} \beta)^T (\mathbf{y} - \mathbf{X} \beta) \end{equation} I know that the ...
0
votes
0answers
12 views

Implicit differentiation(determine derivative of y(x))

I have a function: $$F(x,y) = 2x^4 + 3y^3 +5xy$$ And input $x$ and output $y$ we know that this relation $F(x,y) = 10$ confirms. We know, that this happens when x = 1 and y = 1. By small change of ...
0
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0answers
12 views

Apply Laplace Operator with two functions

Proof that $\Delta u(x,y):=\partial_1\partial_1u(x,y)+\partial_2\partial_2u(x,y)=0$ is valid for $\log(x^2+y^2)$ and $\arctan\left(\frac{x}{y}\right)$. Is it enough to differentiate the functions in ...
0
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1answer
48 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
0
votes
1answer
20 views

computing the directional derivative by definition

The definition of directional derivative: \begin{equation} \frac{\partial f(\mathbf{x})}{\partial \mathbf{v}} = \lim_{h \rightarrow 0} \frac{f(\mathbf{x} + h \mathbf{v})-f(\mathbf{x})}{h}. \end{...
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0answers
57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
0
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2answers
48 views

Integrating product of Dirac delta function and it's derivative

Is the following true: $$ \int \delta(x-a) \frac{\partial^l \delta(x)}{\partial x^l} dx = \frac{\partial^l \delta(x)}{\partial x^l} \Bigg |_{x=a}$$ If not, is there a correct way to evaluate the left ...
38
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8answers
2k views

Does this pattern have anything to do with derivatives?

In 6th grade I was first introduced to the idea of a function in the form of tables. The input would be "n" and the output "$f_n$" would be some modification of the input. I remember finding a pattern ...
0
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2answers
42 views

Antidifferentiation: Stone dropped from $150ft$ rising at $10ft/sec$

A stone is dropped from a balloon when it is $150ft$ above the ground and rising at the rate of $10ft/sec$. How long will it take the stone to strike the ground, and with what velocity does it strike ...
6
votes
4answers
815 views

Why doesn't derivative difference quotient violate the epsilon-delta definition of a limit?

So the difference quotient is defined as: $$\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ So if we take a function such as $f(x)=x^2$ and go through the simplification, we get $$\lim \limits_{h \...
2
votes
4answers
50 views

Antiderrivative of ${d^2 y \over dx^2} = 1-x^2$

At any point $(x,y)$ on a curve, ${d^2 y \over dx^2} = 1-x^2$, and an equation of the tangent line to the curve at the point $(1,1)$ is $y=2-x$. Find an equation of the curve. This is what I've done ...
1
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3answers
42 views

Differentiability of piecewise functions

Check whether the function is differentiable: $$f:\mathbb{R}^2\rightarrow \mathbb{R}$$ $$f= \begin{cases} \frac{x^3-y^3}{x^2+y^2} & (x,y)\neq (0,0) \\ 0 & (x,y) = (0,0) \\ \end{...
0
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0answers
43 views

Calculus: Derivative of a summation and dot product

I'm trying to implement a speed boost to an eye-tracking algorithm (found here: http://www.inb.uni-luebeck.de/publikationen/pdfs/TiBa11b.pdf). I need to take the derivative of the eye-tracking ...
2
votes
1answer
50 views

Neural Network - Why use Derivative

Good Day I am trying to get an understanding of Neural Network. Have gone through few web sites. Came to know the following: 1) One of main objective of neural network is to “predict” based on data....
0
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1answer
43 views

Differential equation without analytic solution - comparative statics

I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value. ...
1
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1answer
49 views

Concavity of function $\sin(x^2)$.

I want to check where the function is convex and where concave. For this I need to calculate the second derivative test: I got $f''(x) = 2*\cos(x^2) -4*x^2(\sin(x^2))$ and this derivative should be ...
0
votes
1answer
24 views

Chain rule confusion(multivariable functions)

I am confused bz the chain rule of multivariable function. I know, that sometimes it is impossible to dot it. For example i have the following functions: $f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^...
0
votes
1answer
80 views

Fréchet differentiability of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$?

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ I want to determine ...
2
votes
1answer
99 views

Why is $\frac{d(x^n)}{d(x)}=nx^{n-1}$

So I was thinking about what I have learnt and I realised that I kind of took the derivative of a function for granted. So I did some research as I wanted to find out how this was discovered and I ...
0
votes
2answers
32 views

Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
0
votes
3answers
29 views

Derivatives: Combining Product & Chain Rules

So I'm working through the material on Khan Academy, and the question is: "Consider the function $f(x) = x^n\ln x$, defined for $x > 0$. Determine, in terms of $n$, the value of $x$ for which $f'...