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

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4
votes
3answers
76 views

Compute $\frac{d^ny}{dx^n}$ if $y = \frac{7}{1-x}$

I am wondering what is $\frac{d^n}{dx^n}$ if $y = \frac{7}{1-x}$ Basically, I understand that this asks for a formula to calculate any derivative of f(x) (correct me if I'm wrong). Is that related to ...
0
votes
0answers
54 views

Does $f'(x) =0$ imply $f(x) = const$ for every domain?

For $f : [a;b] \rightarrow \mathbb{R}$ $f'(x) =0$ implies $f(x) = const$ but what when domain is not an interval?
2
votes
4answers
53 views

Derivative of exponent

Looking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!
5
votes
1answer
78 views

What does it take for a smooth homeomorphism to be a diffeomorphism?

I have an open subset $A$ of $\mathbb{R}^k$ and a subset $B$ of $\mathbb{R}^n$, $n>k$, that are homeomorphic and $f:A\longrightarrow B$ is a smooth homeomorphism between two sets. I'm wondering if ...
-1
votes
2answers
50 views

Find the antiderivative…

Find the complete solution of the given differential equation $${dy \over dx} = {3x \sqrt{1+y^2} \over y}$$ I know how to solve it if the right side didn't contain either $x$ or $y$, but I can't ...
0
votes
1answer
47 views

Find $\frac{dy}{dx}$ when $t=0$ for $\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$

A dot is moving on a grid following this rule: $$\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$$ I need to find $\frac{dy}{dx}$ when $t =0$. It seems like I should use implicit ...
0
votes
1answer
24 views

Partial derivatives and differentiability, continuity

Function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$ has in every $x$ of domain partial derivatives $\frac{\partial f}{\partial x_1}(x) =x_2$, $\frac{\partial f}{\partial x_2}(x) =x_1$, $\frac{\partial ...
0
votes
3answers
42 views

Is $|xy|$ differentiable in $(0,0)$?

Is $f(x,y) = |xy|$ differentiable in $(0,0)$? I have no idea how to approach this problem.
0
votes
0answers
15 views

Transformation of an equation in total derivatives

When I studied calculus many years ago, transforming $f'=df/dt$ into $df=f'\cdot dt$ was always accompanied by words of caution that this is a bit lazy. We used it mainly during integral calculation ...
2
votes
1answer
35 views

Differentiation under the integral sign, where the partial derivative of the integrand is not bounded by a Lebesgue integrable function.

Let $K(t)=\int_1^\infty u(t,x)\ \mathrm{d}x$, where $$u(t,x)=\frac{\cos{tx}}{x^2}\mathbb{1}_{[1,\infty)}(x).$$ I need to show that, for $t>0$, $$\frac{dK}{dt}(t)=\frac{1}{t}\left(K(t)-\cos{t}\right)...
-1
votes
0answers
14 views

Resolution function explicity [closed]

Examine where the equation $f(x,y)=0$ locally by $y=h(x)$ can be resolved. Calculate in all these places $h'(x)$ by implicit differentiation. Enter the resolution function(s) $h(x)$ explicitly if this ...
-3
votes
0answers
35 views

Please prove the following question from isc class12 math book [duplicate]

Let $$y=\sqrt {1-\sin 2x\over 1+\sin 2x}$$ , prove that $${dy\over dx}+\sec^2 (\pi/4-x)=0$$
-5
votes
1answer
111 views

Derivative of $\sqrt {1-\sin 2x\over 1+\sin 2x}$ [closed]

Let $$y=\sqrt {1-\sin 2x\over 1+\sin 2x}$$ , prove that $${dy\over dx}+\sec^2 (\pi/4-x)=0$$
2
votes
0answers
70 views

Any reason not to define a derivative as the average of the derivatives on all sides?

We all know $\operatorname{abs}$ is not differentiable in a classical sense, but one question that's always bothered me is, why not define the derivative as the average derivative in each direction? i....
0
votes
0answers
13 views

Starting from an expression of E(V) and P(V), is there a way of obtaining an expression for E(P)?

I have a function, an equation of state named the Birch-Murnaghan's equation of state, in which the Energy ( $E$ ) is a function of the Volume ($V$), where $E_{0}$, $V_{0}$, $B_{0}$ and $B_{0}^{'}$ ...
1
vote
1answer
46 views

Differentiation under integral sign- Multivariable case problem

Let $f_{\theta}(x,y)=f(x\cos \theta-y\sin \theta,x\sin\theta+y\cos\theta)$, where $f\in C^2(\Bbb{R}^2)$(Is the range necessarily $\Bbb{R}^2$? This is quite ambiguous.) a function with a bounded ...
8
votes
3answers
650 views

Why is/isn't the derivative of a differentiable function continuous?

I am confused about the following Theorem: Let $f: I \to \mathbb{R}^n$, $a \in I$. Then the function $f$ is differentiable in $a$ if and only if there exists a function $\varphi: I \to \mathbb{R}^n$ ...
1
vote
1answer
52 views

A differentiation with first principles question for two variables

I know this question is probably quite easy but it's been some time since I've done any sort of calculus and since a google search failed to turn up anything relevant to this specific question I ...
1
vote
0answers
19 views

Definition of the left and right derivative.

The definition of the derivative is $$g'(a)=\lim \limits_{\delta \rightarrow 0} \frac{g(a+\delta) - g(a)}{\delta}$$ also the left derivative is $$ \lim \limits_{\delta \rightarrow 0^-} \frac{g(a+\...
3
votes
1answer
92 views

How to show for a distribution $T$ and a test function $\varphi,~~T'[\varphi]\equiv -T[\varphi']\;?$

For a generalized function $T,$ we define $$T'[\varphi] ~≡~ −T[φ']~~~~~~\forall φ ∈ \mathcal D(Ω).$$ where $\mathcal D(\Omega)$ denotes the test function space. I'm not getting how they ...
0
votes
1answer
23 views

How to prove that a function only defined for integers (or primes, or multiples of 3, etc.) have a certain derivative?

I would like to know how a function not continuously defined (for a lack of better words) could be proven to have a certain derivative (or, if the word "derivative" isn't appropriate, rate of growth). ...
1
vote
0answers
16 views

Finding the upper derivitive of the compostion of two functions

Let $f$ be defined on $[a,b]$ and g a continuous function defined on $[\alpha , \beta ]$ that is differentiable at $\gamma \in (\alpha, \beta)$ with $g(\gamma)=c\in(a,b)$. Show that if $g'(\gamma)>...
0
votes
1answer
58 views

Differentiation w.r.t. the $\mbox{vec}$ operator

I am stuck at solving the following derivative $$\frac{d \mbox{vec} (X^T X)}{d \mbox{vec} (X)}$$ where $X$ is an $m \times n$ matrix and $\mbox{vec}$ is the vector/stack operator. I have tried using ...
0
votes
0answers
22 views

Do different methods of calculating fractional derivatives have to be equal?

Do different methods of calculating fractional derivatives have to be equal? Or do they sometimes end up differently? An example would be nice, and if possible, an explanation as too why such ...
0
votes
1answer
22 views

Finding a function based on the tangent line

I need your help with this question: The tangent line to the function f(x) at x=1 is y=3x-2. Find f(x) (without using integrals). I know that the derivative at x=1 should be 3, but without more ...
1
vote
1answer
25 views

Derivative of Lattice Laplacian

The lattice Laplacian is defined as, $$ \nabla_L^2x_j \equiv \frac{x_{j+1} - 2x_j + x_{j-1}}{a^2} $$ where the lattice spacing, $a$, is a constant. The derivative, with respect to $x_i$, then gives, ...
0
votes
1answer
38 views

How to prove the following function is convex?

I was working on a problem and it reduced to show that $$f(a)=log\Big(\sum_{i=1}^{r}a^ix_i\Big)~~a>1, x_i>0$$ is convex. I have $$f^{\prime \prime}(a)=\frac{\partial^2f(a)}{\partial a^2}=\frac{[\...
2
votes
1answer
53 views

Is $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth functions of dual argument

Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument? Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs ...
0
votes
1answer
57 views

Can we say that $f(x)$ exactly one common point with the line segment $y=\alpha x$?

Let $f:(0,1) \to \mathbb{R}$ be a continuous decreasing function, and $\alpha \in {R^ + } - \{ 1\} $? Can we say that $f(x)$ exactly one common point with the line segment $y=\alpha x$ such that $...
0
votes
2answers
42 views

Having trouble with partial derivatives

I am having trouble calculating partial derivatives of a simple function. The function is: $$ y(a,b,c)=\frac {0.99821*(a-b)}{c-b} $$ And I need to calculate $ \frac {\partial y}{\partial a} $, $\...
0
votes
2answers
30 views

Please explain this differentiation step

I don't get how they went from line 1 to line 2. Which one is treated as the variable and which the constant? I rearrange line 2 to get $0=\frac{3\varepsilon}{M}-h^3$, but I still cannot see how we ...
4
votes
1answer
62 views

Existance of a function whose derivative of inverse equals the inverse of the derivative

I've been thinking about the calculation of inverse function through Taylor series expansions. My hypothesis was that if we had $$\ f(x) =\sum_{n=0}^\infty \frac{(x-x_0)}{n!}f^{n}(x_0),$$ then $$\ f^{-...
0
votes
1answer
52 views

Directional Derivative defines Descent Direction

Let $f:\mathbb{R}^m \mapsto \mathbb{R}$ be a proper convex function that is not necessarily differentiable and let $x\in\mathbb{R}^n$ be such that $\mathbf{0} \notin \partial f(x)$. I want to prove ...
1
vote
1answer
28 views

Is there a derivative for $|x|$ at $0$ specifically “in the direction” of positive $x$?

I know that $|x|$ is not differentiable at $x=0$ because there is potentially an infinite number of tangent lines going through that point. But let's say we were interested in the motion of an object ...
3
votes
3answers
268 views

Are derivatives always continuous? [duplicate]

I am assuming first off that the derivative exists everywhere on the real number line (or everywhere in whatever set you choose to work in if for some insane reason you drag complex numbers or ...
3
votes
0answers
49 views

Is the $x$-axis a differentiable function? [closed]

Is the $x$-axis a differentiable function?
16
votes
3answers
2k views

Derivative/integral relationship appears to disprove the fundamental theorem of calculus!!!

Consider the floor function: $$f(x) = \lfloor x \rfloor$$ The indefinite integral of f is: $$\int_0^x f(x) dx = x\lfloor x \rfloor - \frac {\lfloor x \rfloor^2 + \lfloor x \rfloor} 2$$ This should ...
1
vote
0answers
23 views

Show that $f$ defined on the interval $(a,b)$ is not differentiable for every point in $E$ with $m(E)=0$

Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for ...
3
votes
2answers
42 views

$2$ points on a curve have a common tangent

Let $2$ points $(x_1,y_1)$ and $(x_2,y_2)$ on the curve $y=x^4-2x^2-x$ have a common tangent line. Find the value of $|x_1|+|x_2|+|y_1|+|y_2|$. It seems to me that I a missing a link and hence the ...
-1
votes
1answer
55 views

Calculus 3 - directional deriviative

I recieved the following question: Calculate the directional deriviative at the point (0,0), of the function: $f(x,y) = x^{2}y + xe^{(x-y)}$ and in a direction that is tangent to the curve: $x^{2}...
0
votes
2answers
28 views

Derivative of function defined by integral of different variable

I have the following exercise which I certainly have gotten no clue about it. Let F(t) be defined: $F(t) = \int_{tan(t)}^{\sqrt{t^2+1}} e^{-tx^2}dx$ What is $F'(0)$? I have no clue about ...
2
votes
0answers
20 views

Sign function identity proof

The signum function is defined by$$sgn(t)=\left\{\begin{matrix}-1, \ t<0\\0, \ t=0 \\ 1, \ t>0 \end{matrix}\right.$$has derivative$$\frac{d}{dt} sign(t) = 2 \delta(t)$$Use this result to show ...
1
vote
0answers
32 views

Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$ f(z) = z^{11} + 4 e^{z + 1} - 2 $$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
1
vote
3answers
34 views

Third derivative of $y=at^2+2bt+c$ and $t=ax^2+bx+c$

If $y=at^2+2bt+c$ and $t=ax^2+bx+c$. Then find $$\frac{d^3y}{dx^3}$$ Now $\frac{dy}{dx}=(2at+2b).(2ax+2b)$ but to proceed further as $\frac{dy}{dx}$ is function of $x,t$
0
votes
0answers
25 views

Successive differentiation

Find the value of $y_n$ for $x = 0 $,when $$ y = e^{(a sin^{-1}(x))}$$. In the my book its already solved the problem is that I don't understand after a certain point ,the steps. After solving we ...
1
vote
2answers
70 views

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$. Then there exists $C$ in $\mathbb R $ such that $|f(x)|\leq C \sqrt |x|$ ...
0
votes
1answer
31 views

Double derivative w.r.t x and y needed

I have the following function $$h(x,y)=\int_{\frac{eaf}{c(1-x)}}^\infty e^{-t-\frac{eagf}{cx(1-y)t}}dt$$ where $a,c,e,f,g$ are constants. I need to find the double derivative w.r.t. $x$ and $y$ i.e. $...
38
votes
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 ...
3
votes
2answers
35 views

Conjecture about Cal 1 derivatives?

Conjecture: Let $F\left(\vec{x}\right) : \Bbb{R}^n \to \Bbb{R}$ Define $g(t) = F(t, t, \dots, t)$ Then $$g^{\prime} (t) = \left(\sum_{i=1}^n \ { \partial F \over \partial x_i}\right)\...
1
vote
1answer
31 views

Repeating/“Periodic” Derivatives? [duplicate]

We know that $Ce^x$ and $0$ are the two functions whose first derivative is equal to itself, but what about derivatives of a higher order? For example, the second derivative of $e^{-x}$ is equal to ...