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

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-1
votes
2answers
17 views

Given $y=(x+1)^{6x}$ use logarithmic differentiation to find $\frac{dy}{dx}$

The final answer I got after working this is $\frac{6(x+1)^{6x}}{x+1}$. But I am not confident in my answer. Is this correct?
3
votes
5answers
130 views

$99$th derivative of $\sin x$

Can someone help me calculate the $99$th derivative of $\sin(x)$? Calculate $f^{(99)}(x) $ for the function $f(x) = \sin(x) $
0
votes
1answer
47 views

Differentiation/ find the derivative

Can anybody please help me with my work? I have to find the differentiate/ find the derivative of these two question: Please HELP!!! $sin^2(cos3x^3)^5 $ $cot^2(x)((x^2)(3cos^3(3x)))^2$
1
vote
4answers
88 views

Antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$

What is the antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$ ? $\displaystyle\int t^2e^{-\frac{1}{2}t^2}\,dt=\displaystyle\int{t}_{}te^{-\frac{1}{2}t^2}\,dt=-te^{-\frac{1}{2}t^2}\Big|_{?}^?+\int ...
0
votes
1answer
14 views

Locate and classify stationary points

Locate and classify as maxima, minima or saddle point the stationary points of the surface given by the equation $$z=(5x+7y-25)e^{-(x^2+xy+y^2)}.$$ Stationary points are the points where the gradient ...
1
vote
1answer
29 views

Jacobian of mapping

Let's say we're in $\mathbb{R}^n \times \mathbb{R}^n$ and we have the identity mapping $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n \times \mathbb{R}^n$, $f(x,y) = (x,y)$. What I want ...
2
votes
1answer
23 views

Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
1
vote
1answer
16 views

Graphing derivatives of non-function equations

Is it possible to graph the derivatives of equations that fail the vertical line test? Such as a circle, a folium of descartes, an asteroid, etc?
0
votes
1answer
19 views

Special Derivative Function

I have a function $f(x)$ which I would like to have the derivative with respect to $x$. How can I get the derivative of the following function with respect to $x$? $$f(x) = \log(1-z^{e^{y^{T}x}})$$
0
votes
4answers
32 views

Find the absolute maximum/minimum values of S(t) where S'(t) is a quartic function with lots of horrible decimal places.

So I have a problem where I'm to find the absolute maximum and minimum values of the following function... $S(t) = -0.00003237t^5 + 0.0009037t^4 - 0.008956t^3 + 0.03629t^2 -0.04458t + 0.4074$ ...
6
votes
4answers
795 views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
1
vote
1answer
24 views

Find speed of an aircraft flying towards an observer

An aircraft is flying towards an observer at an altitude of $2000$ m. When the angle of observation, $z$, (between the ground and the aircraft) is 30 degrees, the rate of change of this angle is $2$ ...
5
votes
3answers
77 views

Limit of $x \ln^2|x|$ when $x\to 0$

I want to evaluate this limit :$$\lim_{x\to 0}x\ln^2|x|$$ I wanted to use L'Hôpital's rule for this: $\lim\limits_{x\to0-}\frac{\ln^2|x|}{\tfrac{1}{x}}$, but I don't know how to differentiate the ...
1
vote
1answer
29 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
1
vote
1answer
11 views

Related rates of change - concentric spheres

Two concentric spheres each have an initial volume of 0. Their radii are increasing at 3mm/s and 5mm/s respectively. Calculate the rate at which the volume between the spheres is changing after 4 ...
1
vote
1answer
25 views

The derivative of $x^TAx$ w.r.t $t$

Suppose $P = x^TAx$ How to find $\frac{dP}{dt}$? if $x' = Bx$ , where $B$ has the same dimension as $A$. How to find the final answer? my answer is: $$\frac{dP}{dt} = 2[(A+A^T)x]x' = ...
0
votes
1answer
22 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
0
votes
2answers
20 views

Show that $f$ is everywhere differentiable and the partials commute

Take the function $$ f(x,y) = \begin{cases}\frac{x^3y -xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}. $$ Show that it is everywhere differentiable and that $D_{1,2}f(0,0)$ ...
2
votes
1answer
21 views

Finding Tangent line for a Graph with the Natural Log

I'm really confused on how my professor did this problem. Any in depth explanation would be awesome. Thanks for your time.
0
votes
2answers
36 views

$f(x)=-4x^2+11126x-62516$. Time and how many

The question is this I've been trying to get my head around this but simply cannot and am hoping you might get me going. Q: The Store is open from $8$ am-$8$ pm every single day. $X$ represents the ...
1
vote
2answers
40 views

How to find the second derivative?

I use this article from Wikipedia to build it in my program. How to find the second derivative in $(x_i, y_i)$ point of this cubic interpolation, if I know other $(x_j, y_j)$ points?
3
votes
1answer
51 views

How does the chain rule work for more than one variable?

I know that that $$\dfrac{d\sqrt{x}}{dt} = \dfrac{d\sqrt{x}}{dx} \dfrac{dx}{dt}$$ In this equation there you only have 1 variable, namely $x$. But why is the following correct?: $$T = \frac{1}{2} ...
0
votes
1answer
30 views

Partial derivatives of $f(x,y)=\sqrt{|xy|}$

$f(x,y)=\sqrt{|xy|}$ First question: How to find $f_x(0,0)$ and $f_y(0,0)$? I have figured out this using definition - Both are $0$. My next question is: How to show that $f_x(0,0)$ and $f_y(0,0)$ ...
2
votes
2answers
78 views

If $f$ is twice differentiable then $f^{-1}$ is twice differentiable

$f:(a,b) \rightarrow (c,d)$ is a bijection and $f$ is differentibale with $f'(x) \neq 0$ for all $x \in (a,b)$, then $f^{-1}$ is also everywhere differentiable. Show that if $f$ is twice ...
1
vote
2answers
50 views

Are these derivatives correct??

Take take the function defined as $$f(x) = \left\{ \begin{array}{ll} exp(\dfrac{-1}{x^{2}}) & \mbox{if } x \neq 0 \\ 0 & \mbox{if } x = 0 \end{array} \right. $$ Now I am asked to check ...
0
votes
0answers
44 views

Question about limit with derivatives

If the limit of $f(x)$ is finite and $f'(x) \ge 0$ for $x \ge 0$. Is the limit of $f'(x)=0$ as $x$ approaches infinity? I'm struggling to dis/prove this question. It's a follow up question to the ...
3
votes
0answers
33 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
1
vote
1answer
15 views

Can Moore–Penrose pseudoinverse solve for underdetermined linear system?

Thanks for reading my thread. I am thinking, many of us know that Moore–Penrose pseudoinverse can solve for overdetermined system $Ax=b$, where $x=(A^TA)^{-1}A^Tb$; for exmplae the linear regression ...
1
vote
1answer
46 views

Showing a function has first and second partials everywhere and continuity of the partials

Let $f \left( \begin{array}{ccc} x \\ y \end{array} \right)= \begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2} & \mbox{if $(x,y) \neq (0,0)$}\\ 0 & \mbox{if $(x,y) = (0,0)$} \end{cases}. $ I ...
1
vote
4answers
33 views

Show limit of $n\log(1+ \frac{x}{n})$ exists

How would I do this question? The fact that $y \rightarrow \log(1+y)$ tells me that: $$\lim_{h \to 0} \frac{\log(1+0+h)-\log(1)}{h}$$ tends to a existing limit. How do I use this for my answer??
1
vote
3answers
26 views

The Notation for Derivatives

"The derivative of a sum is the sum of derivatives" Above theorem can be mathematically expressed as: $$h'(x)=f'(x)+g'(x)$$ where $f(x)$ and $g(x)$ are two differentiable functions. What is the ...
5
votes
0answers
41 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
2
votes
0answers
30 views

Interchange of limiting operations (question from an engineer)

I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details. 1- Given a double integral \begin{equation} \int_{X}\int_Y ...
0
votes
3answers
41 views

Proving Differentiability rigorously

Assume that real function f is differentiable at $x_0$ with $f'(x_0)$ >0. How would one show that there exists a $\delta$>0 such that $$ f(x)>f(x_0) $$ for all x in between $x_0$ and $ x_0 + ...
2
votes
1answer
30 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...
1
vote
1answer
45 views

One-sided Derivative Question

Let's say we define $$D_{+}f(x):=\lim_{h\to 0^+}\frac{f(x+2h)-f(x+h)}{h}$$ to be the "right-handed" derivative. This way the function does not have to exist (or equal what it 'should') at the point ...
1
vote
1answer
19 views

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2)

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2) is _______ I worked out the limit using L'Hospital got a relation ...
2
votes
3answers
159 views

What is defined by rate of change at a single point?

Rate of change measures how fast a process is going when it moves from one point to another. It measures the change of, say, $Y$ when $X$ moves from $X$ to $X + \Delta X$. But my problem arises when ...
0
votes
3answers
38 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
0
votes
0answers
27 views

Intermediate value of the derivative.

Hi all what would the best way be to approach this question? I tried using the hint but I can't seem to formulate an answer for the fist part. Any help for the first and second parts of the question ...
1
vote
1answer
33 views

Differentiation of $u^{T}Su$

I want to differentiate $u^{T}Su$ wrt $u$ where $u$ is $n$ x $1$ and $S$ is $n$ x $n$matrix . So I did the following . Since $u^{T}Su$ is a number , I wrote its expression ie $$ f = ...
0
votes
0answers
8 views

Derivatives of functions defined implicitly

Let $f$ and $g$ be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find the formulas for the partial derivatives of $F$ of first and second order, expressed in terms of derivatives of ...
0
votes
0answers
13 views

Find derivation error and how to prevent it in the future

Problem: Here Calculation: Here. In the yellow spot, I get it wrong. Then I correct it and arrive at the right answer. Question: I'd be glad if someone could find my derivation error and advice me ...
0
votes
0answers
28 views

True or false: differentiation. [closed]

If the function $f(x,y): \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ is differentiable at $(2,-1)$ with a tangent plane such as $z= 2x - 3y + 2$, then the function $g(x,y)= 3x - 2f(x,y) + 5$ is ...
29
votes
4answers
1k views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
1
vote
1answer
27 views

Differentiability of $\sum x^j$

Prove that $\sum x^j$ is differentiable on (-1,1), and $$\frac{d}{dx} \sum x^j = \sum (j+1) x^j$$ I am able to prove that $\sum x^j$ converges uniformly to $\frac{1}{1+x}$. However, how do I get this ...
1
vote
2answers
53 views

Find the derivative of $1/\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}$

(calculus) How can I prove that $$\frac{d}{dx}\frac{1}{\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}}=\frac{-\frac{\displaystyle\pi\sin(\sin(1/x))\cos(1/x)e^{2\pi\cos(\sin(1/x))}}{x^2}+x+\sin x+\cos ...
1
vote
1answer
17 views

Differentiability in two variables - directional derivative & gradient

I have read a chapter about differentiability in two variables. I now have two questions: Why do we need the constraint that $|\vec{u}|=1$ when we calculate the directional derivative? Definition of ...
3
votes
1answer
31 views

Differentiation of $u(t)=\int_0^t h(s,t)ds, \ \forall t \in \mathbb{R}$ with the multivariable chain rule

Problem: Let $h: \mathbb{R}^2 \to \mathbb{R}$ be continuous and differentiable with respect to its second variable, define $u(t)= \displaystyle \int_0^t h(s,t)ds, \ t \in \mathbb{R}$ In an ...
4
votes
1answer
41 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...