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

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0
votes
2answers
45 views

Approximate $f''(3)$ from Table of Values of $f(x)$

Considering the table above, what is the best approximation for $f''(3)$? How would I solve?
3
votes
1answer
57 views

Why do we need to assume continuity in the proof of the chain-rule?

Look at this proof: If $f$ is differentiable at $x$, then it must be continuous there too? Does he then need in the hypothesis that $f$ need to be continuous in the entire interval? What if he just ...
2
votes
0answers
49 views

$\frac{f(x)-f(0)}{g(x)-g(0)}=\frac{f'(\nu(x))}{g'(\nu(x))} $ ,demonstration of the value of the limit: $\lim_{x \to 0^+} \frac{\nu(x)}{x}=0.5 $ [on hold]

Good evening, I thought a lot about this issue. I think I have to apply Lagrange, Taylor. Can someone help me to demonstrate this limit? $$f,g \in C^2 [0,1]: \\ f'(0)g''(0) \ne f''(0) g'(0) \\ ...
2
votes
4answers
167 views

12th order derivative using Leibniz

I don't fully understand the rule of Leibniz and I'm trying to find the 12th order derivative of: $xcos(x)$ How do I find this using the Leibniz rule?
1
vote
1answer
38 views

$\dfrac{dy}{dx} + 0.4y = 3e^{-x}$

calculate $y(3)$ using step size $h=1$ given $y(0)=5$ via euler method solve the differential equation calculate the error between the approximation and actual value of $y(3)$ I got, ...
3
votes
3answers
35 views

$f(x) = x^3-x^2+3x-1$ and $f(1) = 2$, find $g'(2)$ if $g(x) = f^{-1} (x)$

$f(x) = x^3-x^2+3x-1$ and $f(1) = 2$, find $g'(2)$ if $g(x) = f^{-1} (x)$. I'm stuck. Can I get hints only? No anwser please.
1
vote
3answers
67 views

Derivative: Finding the normal line

I have to find the normal line. $$x^2-4x-5 ;(-2,7)$$ So, I have to find the equation of tangent line at $(x_1,y_1)$ which is, $$2x_1-4$$ Now the equation at the (-2,7) is, $$y-7=-8(x+2)$$ ...
1
vote
3answers
38 views

Derivative; The Tangent line $y=x^2 -4x -5 ; (-2,7)$

I was just introduced to derivatives. I have a problem, Find the slope of the tangent line to the graph at the indicated point. The question was, $$y=x^2 -4x -5 ; (-2,7)$$ So the indicated ...
-5
votes
0answers
26 views

differentiation 1 [on hold]

Relate the process of integration to the area under a curve and state the function of the constant of integration. With and illustration, determine the integral of a simple algebraic expression
3
votes
0answers
21 views

Gradient of the distance function

Let $\Omega$ be open, bounded subset of $\mathbb{R}^n$. Let $d(x):=dist(x,\partial\Omega)$ denotes the distance of the point $x\in\Omega$ from the boundary $\partial\Omega$. Define function ...
0
votes
2answers
42 views

Finding inverse of a composite function

Let $f (x) = x^{3}+x$ and $g (x) =x^{3} -x$ for all x. I have to find derivative of $g\circ f^{-1}$ at $x=2$. My textbook did this: $(g \circ f^{-1})' (2) = \lim \limits_{h \to 0} \dfrac{g \circ ...
3
votes
1answer
60 views

Two ways of looking at derivative at $x=0$ of $f (x)= x^{2} \sin1/x $

To show that derivative of $f$ at $x= 0$ exists, where $f(x)= x^{2} \sin1/x$, for $x\ne0$, and $f(0)=0$. I did this by using definition of differentiability, then I found it to be $0$ so it exists ...
4
votes
3answers
69 views

what is meant by $ f ∈ C^{2}[a, b] ?$

What is the meaning of $ f ∈ C^{2}[a, b] ?$ I think it says that $f$ is twice differential on $[a,b]$, isn't it?
-5
votes
0answers
38 views

Can anyone solve question 4 from the Edexcel C3 16th June 2014 paper? [on hold]

The mark scheme and paper are available online. If anyone can solve this question please email me back as I am struggling. The mark scheme doesn't explain how to solve it. Regards.
0
votes
1answer
21 views

Problem with differentiating an identity regarding one-family of diffeomorphisms

I really dont see how the differentiation with respect to t has been done. Before and after the third equation I am ok. Hopefully this is enough information.
24
votes
5answers
2k views

Function whose third derivative is itself.

I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't. Is there any such function? Which one(s)? or How can we prove that there is none? Notes: ...
0
votes
0answers
12 views

vector by vector derivative and the size of final matrix [on hold]

Consider S a 6*1 vector that we are going to derive by O (2*1) then we derive the result again by O . My question is simple:what is the size of the result of derivation operations?
8
votes
8answers
255 views

Why is the differentiation of $e^x$ is $e^x$?

$$\frac{d}{dx} e^x=e^x$$ Please explain simply as I haven't studied the first principle of differentiation yet, but I know the basics of differentiation.
2
votes
1answer
43 views

Prove an identity using differential calculus to a problem connected to fluids

Euler's equation for a incompressible inviscid fluid is $\displaystyle \frac{\partial \textbf{v}_t}{\partial t}+(\textbf{v}_t \cdot \nabla)\textbf{v}_t=-\nabla p_t$ where ...
0
votes
1answer
25 views

what is the partial derivative of sign function? [on hold]

I have a function in the following form. $$ f(x,w)=wx; \quad y = \operatorname{sign}f(x,w). $$ What is the derivative $\frac{\partial y}{\partial w}$ ?
6
votes
1answer
55 views

Prove that $f(x) = a g(x)$

Given two functions $f,g: \mathbb R \to \mathbb R $ such that $g(x) \neq 0 \; \forall x \in \mathbb R$ $f''(x)g(x) = f(x)g''(x) \; \forall x \in \mathbb R$ $f(0)g'(0) = f'(0)g(0)$ Prove that ...
0
votes
1answer
19 views

Showing that $\mathrm{div}_{\mu}X=\frac{\partial X^i}{\partial x^i}$

Let $\mu$ be a non-vanishing $1$-form on $\mathbb{R}^n$. Given a smooth vector field X on $\mathbb{R}^n$, we define the divergence of $X$ wrt $\mu$, denoted by $\mathrm{div}_{\mu}X$, by $L_X \mu = ...
8
votes
3answers
192 views

How can I check if my derivative for an implicit function is correct?

For explicit functions I can calculate the derivative at a certian point using the original function: $$\frac{f(1+0.1) - f(1)}{0.1}$$ And then use $\frac{d}{dx}f(1)$ to check if the function is ...
1
vote
3answers
86 views

function such that $f(x\cdot t)=f(x)g(t)$

Let $E$ be the set of tuples of continuous functions $f,g:\mathbb{R}^*_+\rightarrow\mathbb{R}$ s.t. $f,g$ are never $0$ and $\forall x,t>0,f(x\cdot t)=f(x)g(t)$. I need to show that ...
2
votes
3answers
26 views

Minimum value of a differentiable function at some point

Let $f(x)$ be differentiable for all $x\in \mathbb{R}$ and let $f(0)=2$ and $f^\prime(x)\leq -2$. How could i find the minimum value of $f(-1).$
0
votes
1answer
30 views

Show that if Frechet (d) derivative is exist this implies that Gateaux derivative is exist.

Show that if Frechet (d) derivative is exist this implies that Gateaux (D) derivative is exist , and D f(x)= d f(x), and show that the opposite is not true always and why? my solution suppose that ...
0
votes
1answer
40 views

Derivative of a matrix with respect to a scalar

I would really appreciate some help in finding the partial derivative of the following with respect to $\psi$: \begin{equation} \mbox{trace}((XX^\top)^{-\psi}V) \end{equation} Here, $\psi \in [0,1] ...
2
votes
2answers
37 views

Differentiability of a function satisfying a condition on a bounded interval

Let $\displaystyle f(x)$ satisfies the condition $|f(x)|\leq 1-\cos x$ on the interval $-\frac{\pi}{2}\leq x\leq \frac{\pi}{2}$. Prove that $f(x)$ is differentiable at $x=0$ and find $f^\prime(0)$. ...
1
vote
0answers
16 views

A question about solving directional derivatives

The question comes from the paper ``Regression Quantiles'' by Roger Koenker and Gilbert Bassett(1978). $0< \theta <1$. Define $\psi(b;\theta,y,X)=\sum^{T}_{t=1}[\theta-1/2+1/2 \; ...
3
votes
2answers
62 views

if $f(z),\overline {f(z)}$ are analytic then they are constant

I'm trying to prove this "theorem": if $f(z),\overline {f(z)}$ are analytic in some open set $\Omega \subseteq \mathbb C$, then $f(z)$ is a constant. Hint: Use Cauchy-Riemann equations to show that ...
1
vote
0answers
35 views

Show that $\Phi^*_t\mu=\mu \iff \mathrm{div}_{\mu}X=0$.

Let $\mu$ be a non-vanishing $1$-form on $\mathbb{R}^n$. Given a smooth vector field X on $\mathbb{R}^n$, we define the divergence of $X$ wrt $\mu$, denoted by $\mathrm{div}_{\mu}X$, by $L_X \mu = ...
1
vote
1answer
23 views

Sign of differentiable function near critical point

Background : We know that Brownian path oscillates infinitely often changing signs in any neighbourhood of $0$. I was trying to understand if this property holds because that Brownian paths are not ...
0
votes
1answer
26 views

Let $y = 2au,\;$ and $\;x=a(u)^2.\;\;$ Find $\,\frac{dy}{dx}\,$ [on hold]

Let $y = 2au,\;$ and $\;x=a(u)^2.\;\;$ Find $\,\dfrac{dy}{dx}\,$ at $u=\dfrac 1\pi$
5
votes
5answers
117 views

$f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.

I'll start with the precise statement of the problem: Suppose that $f:[0,b]\to\mathbb{R}$ is differentiable, $\,f(0)=0$, and that there exists a real number $K\geq 0$ such that ...
2
votes
1answer
25 views

Power series for inverse of truncated power series of $e$

Let $T_n(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$. I'm looking at the function $$f(x)=\frac{1}{T_n(x)}$$ and I would like to find the power series of this particular function. I know I can use Maclaurin ...
2
votes
1answer
30 views

derivative of implicit function

supposed I have a function which is $q_2f_2(δq_1+q_2)$ I want to know the second derivative of the function w.r.t $q_2$ Firstly, I took the first derivative w.r.t to $q_2$ and I got the result as ...
0
votes
1answer
41 views

Find a function $f$ on $\mathbb{R}^3$ such that $A^*df=0$

Let $A : \mathbb{R}^2 = \{(u,v)\} \rightarrow \mathbb{R}^3=\{(x,y,z)\}$ be given by $A(u,v)=(u,v,F(u,v))$ Find a function $f$ on $\mathbb{R}^3$ such that $A^*df=0$. I have tried ...
0
votes
1answer
33 views

Limit on a continuous differential equation

Let $f$ be a continuously differentiable function and let $L=\lim_{x\to\infty}(f(x)+f'(x))$ be finite. Does this imply that if $$\lim_{x\to\infty} f'(x)$$ exists, then it is equal to $0$?
5
votes
2answers
46 views

Sufficient conditions to have $f' = O(f(x)/x)$.

Suppose $f$ a nonnegative real-valued function, non-decreasing, $O(x^m)$ for some $m \in \mathbb{Z}_{\geqslant 0}$ and $C^1$, with $f'$ being monotonic and nonnegative. Are this sufficient conditions ...
2
votes
0answers
20 views

Number of terms in $f'$ as a polynomial of the number of terms in $f$

Take an elementary function $f$. The derivative $f'$ is an elementary function as well. Let $G(f)$ be the minimum number of terms required to express $f$ as a formula comprising a finite number of ...
7
votes
2answers
80 views

If f is differentiable on (a,b) continuous at a and f has bounded derivative must f be right differentiable at a?

If $f$ is differentiable on $(a,b)$ continuous at a, and $f$ has bounded derivative, must $f$ be right differentiable at $a$? In case answer to previous question is true, is the statement still ...
0
votes
0answers
36 views

Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
1
vote
3answers
97 views

What's wrong with this derivative for $\left(1−\frac{1}{x}\right)^x$?

I asked a question about how to differentiate $(1−1/x)^x$ before, for $x>1$. The derivative I was told is ...
0
votes
2answers
61 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
-3
votes
2answers
50 views

Consider the function with its first and second derivative help please [duplicate]

$$f(x)=\frac{4x^2}{x^2+3} $$ $$f'(x)=\frac{24x}{(x^2+3)^2} $$ $$f''(x)=\frac{72(1-x^2)}{(x^2+3)^3}$$ a)What are the critical numbers(if any)? b)On what intervals is the function increasing and on ...
2
votes
2answers
61 views

20th derivative of a rational function

I could not find the 20th derivative of the function below : $$f(x) = \frac{2x}{x^2 - 4}$$ I have taken 1st and 2nd derivatives but I could not succeed at generalizing the derivative function.
0
votes
0answers
14 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
2
votes
1answer
49 views

Application of Rolle's Theorem and differentiation

Suppose $f: \mathbb{R}\rightarrow \mathbb{R}$ is differentiable with $f(0)=f(1)=0$ and $\{x:f'(x)=0\}\subset \{x:f(x)=0\}$. Show that $f(x)=0$ for all $x\in [0,1]$. My Work: By Rolle's Theorem ...
1
vote
3answers
60 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
1
vote
1answer
37 views

How do I prove this derivation?

I hope you can help me with this one because I seem to not quiet get a start here :/ Lets say we got a $b\in\mathbb{R}_{\gt 0}$ and a $y\in\mathbb{R}$ and we define $b^y:=\exp\left(\ln b \cdot ...