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

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2
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5answers
93 views

Derivative of $ h(t)= \sin (\cos^{-1}t$)?

Can someone please explain the steps/rules I need to preform to find the derivative of $h(t)= \sin (\cos^{-1}t)$? I tried to used the product rule, and realized it was obviously a failure. Using ...
1
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0answers
34 views

prove that if f is $C^1$ (meaning the derivative $f´$ is continuous) then it can be represented as the sum of an increasing and a decreasing function

prove that if f is $C^1$ (meaning the derivative $f´$ is continuous) then it can be represented as the sum of an increasing and a decreasing function I can´t find any solution for this problem ...
0
votes
1answer
64 views

Critical points and absolute extreme values on given interval

I'm back with a question! I am working on a homework problem and I got stuck. I'm asked to 1.find the critical points of f on the given interval 2. fine the absolute extreme values of f on the given ...
1
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1answer
82 views

Strictly monotone real function: stationary point, non-differentiable point

If we have a real function $f$ that is strictly increasing (or strictly decreasing), what can we say about measure and cardinality of stationary points/points with no derivative. In particular: -Is ...
0
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1answer
35 views

Differential equation, symmetric about 0?

Solving the following numerically (with different values of $u(-1)$) $(2-\cos(\pi x))u''(t) + u(t) = 1$ and $u(-1) = u(1)$ the solutions seem to be symmetric about $0$. Is it true in general (ie no ...
0
votes
2answers
42 views

Find derivative of function

I need help in finding the derivative. I don't even know where to begin with it. I'm learning chain rule in school and do not see how I can apply that here. $$ f(x)=\left(\dfrac {x+1}{x^2+8}\right)^6 ...
0
votes
0answers
42 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
1
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2answers
39 views

Derivatives of Logarithmic function

Determine $f'(x)$ for $f(x) = ln(x + \sqrt{x^2 + 1})$ My handbook has the answer as $\displaystyle\frac{1}{\sqrt{x^2 + 1}}$ with no steps on how they got there. I tried to get there, but somewhere I ...
0
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1answer
47 views

if $f'''(x)$ is continuous everywhere and $\lim_{x \to 0}(1+x+ \frac{f(x)}{x})^{1/x}=e^3$ Compute $f''(0)$

if $f'''(x)$ is continuous everywhere and $$\lim_{x \to 0}(1+x+ \frac{f(x)}{x})^{1/x}=e^3$$ Compute $f''(0)$ The limit equals to $$\begin{align} \lim_{x \to 0} \frac{\log(1+x+ ...
1
vote
1answer
23 views

Partial derivative identity help

Please can some one give me the proof for this, ( I think y and x can be written as parametric equations in terms of C)(∂y/∂x) = (∂y/∂c) / (∂x/∂c))
0
votes
3answers
141 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
0
votes
0answers
48 views

Taylor series like polynomials

Let $U$ be an open subset of $R^n$ and $f:U\rightarrow \mathbb{R}$ a function and $x\in U$ such that in a small neighbourhood of $x$ and for $\epsilon \in \mathbb{R^b}$ sufficiently small we have the ...
1
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1answer
55 views

Does the little-oh relation remain if $f(x)$ and $g(x)$ both integrate or differentiate?

Give two functions $f$ and $g$ with derivatives in some interval containing 0,where $g$ is positive.Assume also $f(x)=o(g(x))$ as $x \to 0$. Prove or disprove each of the following statements: ...
0
votes
0answers
46 views

Covariant Derivatives of contravariant vector in curvilinear coordinates

$$D_mA^p = \partial_mA^p + \Gamma^p_{mn} A^n$$ so $$D_kD_mA^p = D_k(\partial_mA^p + \Gamma^p_{mn} A^n)$$ $$D_kD_mA^p = \partial_k(\partial_mA^p + \Gamma^p_{mn}A^n) + \Gamma^p_{kl}(\partial_mA^l + ...
2
votes
1answer
61 views

How can I complete this proof?

I have finished all but the last line. Thanks in advance for any help. Proof statement: If $X$ and $Y$ are Banach spaces and $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ are both differentiable, then ...
1
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0answers
21 views

Is there anything wrong with this line of reasoning?

Proof statement: If $f:X \rightarrow Y$ is a bounded linear map, then $Df(x)=f$ for all $x \in X$, where $X$ and $Y$ are Banach spaces. Proof: Consider $$\lim_{h\rightarrow ...
1
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1answer
70 views

Bounded Jacobian implies uniform continuity

I am trying to solve the following problems but I am not sure what the difference between the 2 problems is. 1) Prove that is $U = B_r(x)$ (open ball centered at $x$ with radius $r>0$) is an open ...
0
votes
4answers
52 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
2
votes
1answer
80 views

Does the derivative of a bounded smooth monotone function have a limit at infinity?

Let $f \in C^1(\mathbb{R})$ a monotonic function such that $$\lim_{x \to \infty} f(x) = m \in \mathbb{R}$$ Does this imply $\displaystyle\lim_{x \to \infty} f'(x) = 0$? If so, can the hypothesis be ...
2
votes
3answers
95 views

Why are there so many notations for differentiation?

There are so many notations for differentiation. Some of them are: $$ f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad D f(x)\qquad y^\prime\qquad D_x f(x) $$ Why ...
-1
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1answer
85 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
0
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1answer
33 views

Finding the derivative of analytic polynomials

I have just started studying complex analysis and i am stuck with one question. My book says, the derivative of an analytic polynomial with respect to $z$ is equal to the partial derivative of that ...
1
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1answer
66 views

$f$ continuous, monotone, what do we know about differentiability?

I am interested in knowing what we can say in general about when a continuous function $f:\mathbb{R} \to \mathbb{R}$ is differentiable. To my mind, there are various ways a continuous function can ...
0
votes
3answers
116 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
1
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3answers
73 views

complex analysis: If $f$ is analytic and $\operatorname{Re}f(z) = \operatorname{Re}f(z+1)$ then $Im\;f(z) - Im\;f(z+1)$ is a constant

I am having trouble deciphering the reason behind a line in a complex analysis textbook (Complex made Simple by Ullrich, page 360 5 lines down in Proof of Theorem B, for those who are interested). ...
0
votes
2answers
38 views

Numerical Analysis: Given a function and successive derivatives at one point, what's the value of the function at another point?

Example of an exercise I'm trying to solve: Find the value of $f ( 4)$ given that $f (6 )=350 , f ' (6 )=87 , f'' (6 )=30 , f ''' (6 )=4$ and all other higher derivatives of $f (x) at x=6$ are zero. ...
0
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1answer
36 views

Infinite differentiability at zero of function with property $f(x)=o(x^n)$ $\forall n\in\mathbb{N}_+$

Is it true that the differentiable function $f:\mathbb R\to\mathbb R$ such that $f(x)=o(x^n)$ for every positive $n$ is infinitely differentiable at $0$? Given the differentiability constraint, it ...
0
votes
1answer
60 views

Does $f(x)=o(x^n) \forall n$ imply $f^{(n)}(0)=0$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f\in C^\infty$ and $f(x)=o(x^n)$ when $x\to 0$, for every $n\in\mathbb{N}$. Is it true that $f^{(n)}(0)=0$ for every $n\in\mathbb{N}$?
0
votes
1answer
65 views

Check my answer - Differential of $P(A)=\det(A^{-1}-A)$

We are asked to find the differential of $P: GL_n(\mathbb R) \to \mathbb R$, $P(A)=\det(A^{-1}-A)$ and show it is differentiable. If we define $f(A)=\det(A)$ and $g(A)=A^{-1}-A$ then it is clear ...
0
votes
5answers
94 views

Why is $f(x)=|x|$ not differentiable?

Consider the function $f(x)=|x|$, I know that $f$ is not differentiable at $x=0$, but still, when you try to differentiate $f(x)=\sqrt{x^2}$ (which is exactly the same), you get: ...
1
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1answer
71 views

chain rule: $g(x)=(2ra^{rx}+n)^p$

looking for my mistake and I can't find it. here is the layout: $g(x)=(2ra^{rx}+n)^p$ $p(2ra^{rx}+n)^{p-1} (2ra^{rx}+n)'$ $p(2ra^{rx}+n)^{p-1} (2ra^{rx})'+(n)'$ $p(2ra^{rx}+n)^{p-1} (n)'+[(2)' ...
1
vote
2answers
112 views

Chain rule: $f(t)=e^{t\sin2t}$

I'm missing a step somewhere on this: $$f(t)=e^{t\sin2t}.$$ I know I have to use the chain rule but I'm getting tripped up. here are my steps: $\frac d{dx}(e^{t\sin2t})$ $e^{t\sin2t} \frac ...
1
vote
1answer
38 views

Geometrical Calculus - Mini-Max Problem

Two vehicles are heading for a crossroad (point C) and intend to pass straight through. Vehicle A is $100$ km due North travelling at $80$ km/hr towards C Vehicle B is $150$ km due East travelling at ...
0
votes
1answer
23 views

Derivative of $5+ 10e^{-t}\sin(2t-30)$?

For the derivative of $5+ 10e^{-t}\sin(2t-30)$ I am getting this result: $$ -20e^{-t}\sin(2t-30) + 10e^{-t}\cos t2t, $$ BUT my textbook says the answer is: $$ 22.36e^{-t}\sin(2t+86.565). $$ Could ...
3
votes
1answer
53 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
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2answers
107 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
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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
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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
77 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
votes
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
37 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
votes
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 ...
1
vote
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 ...