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

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3
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1answer
52 views

Continuous differentiation for polynomials?

Has this concept been explored & if so what name does it go by? Taking a simple polynomial & its derivatives: $$y = x^3 + x ^ 2 + x + 1$$ $$\frac{dy}{dx} = 3x^2 + 2x + 1$$ ...
0
votes
1answer
34 views

Prove something that is differentiable

The question states If g(x) is differentiable, then for any positive integer $n$, $(g(x))^n$ is differentiable and $\frac d{dx}$$(g(x))^n=(g(x))^{n-1}g'(x). $ Where does the continuity of g enter ...
1
vote
2answers
105 views

How to prove that $\frac{d}{dx}\sin(x)=\cos(x)$

I have to prove that $\dfrac{d}{dx}\sin(x)=\cos(x)$. I used the definition of a derivative: $$\dfrac{d}{dx}f(x)=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$$ $$\dfrac{d}{dx}\sin(x)=\lim\limits_{h\to ...
1
vote
3answers
30 views

Notation for function compositions/derivatives

When given $(f \circ g)'(0)$, does it mean to compose the 2 functions first, then take the derivative of the composed functions and evaluate it at $0$, or take the derivative of $g$ first and evaluate ...
0
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1answer
32 views

Path derivative

Let $\vec y$ be a vector that represents the path of a particle through space. If we define $w$ as the length of the path, would it be correct to say that $\displaystyle \frac{d\vec y}{dw}$ evaluated ...
0
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1answer
25 views

I have a question regarding the relationship between tan(x) and sec(x).

This is a question that has been on my mind for sometime, and I'm getting two separate and contradictory answers to it. If $\tan x = 1$, then what will be the value of $\sec^2 x$? Now, one relation ...
0
votes
1answer
24 views

Finding directional derivatives that exist

Let $$g(x,y,z)= \begin{cases} \frac{xy+xz+yz}{\sqrt{x^2+y^2+z^2}}, & \text{if } xi+yj+zk \neq 0 \\ 0, & \text{if } xi+yj+zk = 0 \\ \end{cases} $$ Use the definition of the directional ...
3
votes
1answer
76 views

Prove that a function is differentiable using the limit definition

Use the definition of the derivative to prove that $f(x,y)=xy$ is differentiable. So we have: $$\lim_{h \to 0} \frac{||f(x_0 + h) - f(x_0) - J(h)||}{||h||} = 0$$ We find the partial derivatives which ...
0
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1answer
23 views

A word problem, selling cakes and finding the maximum

A school class is saving money for a classtrip and therefore sell cakes. The function $f(x)=x(x-25)(x-15)$ describes how much money the class saves in total for selling cakes. f(x) is the total ...
1
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3answers
36 views

Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
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0answers
29 views

if f is a continuous class 1 function then it can be expressed by the sum of an increasinf function and a decreasing function

prove that if: f is a continuous class 1 function on $[a,b]$ then it can be expressed by the sum of an increasing function and a decreasing function I don´t know where to start my demonstration, I ...
1
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1answer
37 views

How to take derivative of sums of absolute values

Take the derivative of $f(m) = \sum_i | x_i - m |$. I've been told that derivative of each term is +1 or -1. How do you show that?
0
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2answers
31 views

Help with differentiation

So I have a practice question.. I am reading material from a calculus book, and I am studying derivatives and differentiation with functions. One of the questions on a practice sheet I have is similar ...
0
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0answers
53 views

Step by Step explanation of derivative of a matrix

I'm working on a proof that requires me to simplify the derivative of a positive definite matrix. I'm relatively new to matrix calculus so I have been searching the internet for a good example. I ...
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5answers
78 views

Solving $\frac{x}{1-x}$ using definition of derivative

I was trying to find the equation of the tangent line for this function. I solved this using the quotient rule and got $\frac{1}{(x-1)^2}$ but I can't produce the same result using definition of ...
1
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1answer
30 views

Equation of tangent line for $y' = \frac{x}{(1-x)^2}$ at point $(0,0)$

I tried to solve this by plugging zero into x the $x$ values and I end up getting $\frac{0}{1}$, which obviously is $0$. From there I multiply out and get all zeros. What am I doing wrong? More ...
0
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0answers
59 views

Find the mistake in my solution for the derivative of $x^x$

We can easily find the correct derivative of $x^x$ with logarithmic differentiation as follows. $$ \begin{eqnarray*} y & = & x^{x}\\ \ln(y) & = & x\ln(x)\\ \dfrac{dy}{dx}\dfrac{1}{y} ...
1
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0answers
28 views

Derivative of double integral over Region

If one has a function $f_{1}(x)$, which needs to be integrated over a region $x>b$, where $b$ is some one-dimensional boundary, one writes: $$ \int_{b}^\infty dx f_{1}(x) $$ One can, then, take ...
0
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1answer
44 views

Din derivatives and fundamental theorem of calculus

I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function ...
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0answers
35 views

“Painless Conjugate Gradient”: alpha minimizes f when the directional derivative $\frac{df(x1)}{d\alpha} = 0$

I am reading the "Painless Conjugate Gradient Method" http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf On page 6, after equation 9, the author states "From basic calculus, $\alpha$ ...
1
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1answer
36 views

differential (Jacobi Matrix) of $f(A)=A^2$ where $A$ is a matrix - check my answer

I just want a quick verification that what I did here is correct: let $f(A)=A^2$ where $A$ is a n by n matrix with real entries. then $$D_f(A)=\lim_{t \to 0} \frac{f(A+tA)-f(A)}{t} = \lim_{t \to 0} ...
0
votes
1answer
113 views

The normal line intersects a curve at two points. What is the other point?

The line that is normal to the curve x^2 + xy - 2y^2 = 0 at (4,4) intersects the curve at what other point? I can not find an example of how to do this equation. Can someone help me out?
2
votes
3answers
77 views

How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule?

How do I prove that the derivative of $\sin$ is $\cos$ without resorting to L'Hôpital's rule (circular logic)? This part is easy: $$ \begin{align*} \sin'(x) &= \lim_{\Delta x \to 0} ...
0
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1answer
26 views

Integral help needed

I'm a Calc I student trying to understand my homework. What are the steps to figure out this below? The correct answer is 2/x^5. It would be helpful if explained in a method a Calc I student would ...
0
votes
2answers
37 views

Maximum and Minimum Values of the function

What will be the maximum and minimum value of the following function, $f(x,y)=3x+4y$ in the region $0\le x \le1$, $-1\le y \le1 $
0
votes
1answer
31 views

Uniform convergence to the derivative

Suppose $\phi \in C^{\infty}(\mathbb{R})$. Then is it true that $$ \frac{\phi(x+h)-\phi(x)}{h} \to \phi'(x) \quad (h \to 0) $$ uniformly on $\mathbb{R}$? It seems to me like this is trivially the case ...
0
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1answer
60 views

Derivative of exponential functions

Can anyone present an intuitive reason for why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves? i.e. $$ y = e^{x^2}\\ ...
0
votes
1answer
20 views

Defining the function near some points.

How we can show that , $$\ln(x+2y)+32x^3y^2=\frac{1}{4}$$ defines $y$ as a function of $x$ near the points $\displaystyle\left(\frac{1}{2},\frac{1}{4}\right)$ and calculate $y'(1/2).$
0
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4answers
58 views

conjugate function prove derivative

If I know that $f(z)$ is differentiable at $z_0$, $z_0 = x_0 + iy_0$. How do I prove that $g(z) = \overline{f(\overline{z})}$ is differentiable at $\overline z_0$?
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3answers
65 views

Derivative of a function.

How to show that the function $f(x,y)=x^2 y$ is differentiable at $(1,-1)$ by using the defintion and also find the tangent plane for the surface $z=f(x,y)$ at $(1,-1)$
0
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0answers
93 views

Second derivative wrt complex parameter

I'm facing an estimation problem and I need to calculate the Cramer-Rao Lower Bound of an estimator. So I have 2 unknown parameters: the amplitude of the signal $A$ and its direction of arrival $u$. ...
1
vote
1answer
45 views

Infinitesimals in gradients

Take the function $y(\vec v)$ such that $y:\mathbb R^n\to\mathbb R$. Given it's gradient $\nabla y = \left(\frac{\partial y}{\partial v_1},\cdots,\frac{\partial y}{\partial v_n}\right)$, it is ...
2
votes
0answers
48 views

show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...
2
votes
1answer
54 views

Question on a derivative on a Hilbert space

I have this functional $J(u)=\frac12 \|u\|^2+\int_0^1 F(t,Ku(t))dt$ where $F(t,u)=\int_0^u f(t,\xi) d\xi$,$\displaystyle Ku(t)=\int_0^1 G(t,s)u(s) ds$ with $G(t,s)=\begin{cases} s(1-t),&0\leq s ...
1
vote
1answer
34 views

Let f: $\mathbb{R}^2 \mapsto \mathbb{R}^2$ be a linear function, proof about $f$ and directional derivative

I have this $f$ that is linear and I want to show that for any $a,v \in \mathbb{R}^2$ $f(\begin{matrix} a_1 + v_1 \\ a_2 + v_2 \\ \end{matrix})$ = $f(\begin{matrix} a_1 \\ a_2 \\ \end{matrix}) + ...
0
votes
1answer
25 views

show that $\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' (t)\rangle$

Let $\gamma,\eta:[a,b]\to \mathbb R^n$ be continuous, differentiable, curves. show that $$\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' ...
0
votes
1answer
18 views

derivative theory problem

Let f be a derivable function in its domain D and $x_0$ an interior point of D. Prove that: if $$f´(x)>g´(x) \forall x$$ and $$f(x_0)=g(x_0) \Rightarrow f(x)>g(x) \forall x>x_0 and ...
0
votes
0answers
30 views

Proving that derivative of a bounded linear map (at a point) is the map itself

I have been struggling to prove a claim. How can I show that If $f:X \rightarrow Y$ is a bounded linear map, then $Df(x)=f$ for all $x \in X$? Attempt: $f:X \rightarrow Y$ is a bounded linear map ...
0
votes
1answer
35 views

Find all solutions to a particular differential equation

Find all solutions on ${R}$ of the differential equation $ y' = 3|y|^ \frac{2}{3} $ I believe I need to use separation of variables, but it will only work if the initial condition is nonzero. ...
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1answer
49 views

Which line is the antiderivative and why?

The graph of a function $f$ is shown. Which graph is an antiderivative of f and why? This should be easy but I keep second guessing myself so I thought I'd check with you magnificent people. I'm ...
1
vote
1answer
49 views

Check my answer - Finding the jacobi matrix of a function

We are given $f: \mathbb R^n \to \mathbb R^n$ such that: $0 \neq x \in \mathbb R^n$, $f(x)=\frac{x}{|x|}$, where $|x| = \sqrt {x_1^2 +x_2^2+...+x_n^2}$ Find the jacobi matrix (the differential ...
0
votes
1answer
29 views

introductory calculus - Help me find a function with a few properties

I was asked to find a function $f: \mathbb R^2 \to \mathbb R$ such that: 1) $f$ is continuous at $(0,0)$. 2) $f$ has directional derivatives at $(0,0)$ (does this mean $f$ is differentiable at ...
3
votes
3answers
58 views

Does continuity imply existence of one sided derivatives?

From what I understand a derivative may not exist at a given point if the function is not continuous or the right and left side derivatives are not equal. Does that imply that if a function is ...
2
votes
1answer
64 views

Diferentiable function with non-differentiable inverse

Is it possible to define bijective function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable in a point $x_0$ such that $f'(x_0) \ne 0$, but $f^{-1}$ is not differentiable in $f(x_0)$? I think ...
1
vote
1answer
49 views

Mean Value related problem.

I'm working on a function $f : \left[a,a+h\right] \rightarrow \mathbb{R}$. I know that $f$ satisfies the conditions of the Mean Value Theorem thus I have $\theta \in \left(0,1\right)$ such that ...
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2answers
61 views

Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$

Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$ I could not in any way understand how to approach this problem. I think I will be able ...
0
votes
2answers
44 views

how to get $2/(t^2 + 1)$ as the derivative for Sin(theta) when $\tan(\theta/2) = t$

If $\sin \theta = \frac{2t}{1 + t^2}$ How do you get $d\theta = \frac{2}{1 + t^2}$ If you differentiate by quotient rule you get $\frac{2(1 - t^2)}{(1+t^2)^2}$ It is part of the solution to ...
0
votes
1answer
67 views

why $dydy=dy^2$ instead of $d^2y^2$

The first derivative is defined by $$\frac{dx}{dy} = \lim_{\Delta y\to 0}\frac{x(y+\Delta y)-x(y)}{(y+\Delta y) - y}$$ The second derivative: $$\frac{d(\frac{dx}{dy})}{dy} = \frac{d}{dy}\frac{dx}{dy} ...
0
votes
3answers
54 views

Quotient vs Product Rule

You are asked to differentiate $$ y = \frac{x - 1}{x + 1} $$ Looking at the question, I'm thinking I could solve this question using the product rule by making $\tfrac{1}{x + 1}$ into $(x + 1)^{-1}$. ...
0
votes
1answer
23 views

Derivative of restriction of $f$ to parametrized line

Given $f : \mathbb R^n\rightarrow \mathbb R$, if we parametrize $f$ as follows $\phi(\theta)=f(a+\theta(x-a))$, $a\in \mathbb R^n$ is a constant vector. What will be $\phi'(\theta)$ ? I ...