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

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65 views

Suppose that $f$ is a differentiable function such that $f(g(x)) = x$ and $f^\prime(x) = 1+ [f(x)]^2$. Show that $g^\prime(x) = \frac{1}{1+x^2}$.

Suppose that $f$ is a differentiable function such that $f(g(x)) = x$ and $f^\prime(x) = 1 + [f(x)]^2$. Show that $g^\prime(x) = \dfrac{1}{1+x^2}$. $$ f(g(x)) = x \implies f = g^{[-1]} $$ I have ...
1
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1answer
45 views

Curves and tangent vectors in a manifold setting

Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions) Def: A smooth curve in $M$ is a map $\gamma: I \rightarrow ...
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0answers
50 views

derivative of matrix exponential

How to express $\nabla\exp(i\theta(\mathbf{r}))$ in terms of $\nabla\theta(\mathbf{r})$ where $\theta$ is a Hermitian matrix of $n\times n$? Here $\nabla$ means calculating the gradient wrt. ...
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0answers
16 views

Fit derivative to a set of points

Let's say I have a set of discrete values $X = {x_1, x_2, x_3, ..., x_n}$ from the sampling at a rate $f_s$ of a continuous function. I scale some values in $X$ (in a different manner for each one), ...
1
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1answer
32 views

Derivative of a Radial Basis Function $\nabla_x \frac{1}{1+\|x - y\|^d}$

I want to calculate the following derivative (with respect to x): $\nabla_x \frac{1}{1+\|x - y\|_2^d}$ where, $x,y \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector.
0
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1answer
22 views

Does there exists a proof that any divergentless tensor can be decomposed into the sum of divergentless symmetric and antisymmetric tensors?

A friend and I attempted to work out the proof on the board that any divergentless asymmetric tensor can be written as the sum of divergentless symmetric and antisymmetric tensors. We wrote down the ...
2
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1answer
69 views

Showing that $\lim_{x \to 0}\frac{f(x)}{g(x)} = \frac{f'(0)}{g'(0)}$.

If $f$ and $g$ are differentiable functions with $f(0) = g(0) = 0$ and $g'(0) \neq 0$, show that $\lim_{x \to 0}\frac{f(x)}{g(x)} = \frac{f'(0)}{g'(0)}$. I consider that perhaps: $$ \begin{align} \\ ...
0
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1answer
42 views

$f(x) = \ln( \exp(x/2) + \exp(-x/2) )$ is a concave function of $x^2$

In one paper I see this sentence. Not quite sure how to verify it, I take the form $f(x^2)$, and take the second derivative w.r.t. $x^2$, using the chain rule. But at last I found an expression that ...
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2answers
36 views

Proof: f total differentiable then f continuous

I'd like to show that if $f: O \subseteq \mathbb{R} n \to \mathbb{R}m$ is differentiable in $x_o \in O$, then $f$ is continuous in $x_o$. My idea: If $f$ is (total) differentiable in $x_o$ then ...
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1answer
56 views

Optimizing for the minimum relative distance in a given situation?

I have primarily been working on this problem for quite some time now; the level of the problem is introductory calculus w/ optimization problems. The situation is as follows: Ship A sails due ...
2
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1answer
58 views

Why we cannot simplify $\partial x$?

First consider the formula: $$\frac{df}{dt}=\frac{df}{dx}\frac{dx}{dt}$$ As we can see, $dx$ can be simplified from the RHS to get the LHS. This can be explained like this: define $y=x'(c)(t-c)+x(c)$ ...
1
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1answer
18 views

$g$ is differentiable and $g'(y)=\int_{\mathbb{R}}ixf(x)e^{iyx}dm(x)$

Let $f \in \mathcal{L}(\mathbb{R},\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ measurable Lebesgue. Asumme that $x\to f(x)$ is measurable. For $y \in \mathbb{R}$ define: ...
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0answers
56 views

Show that $f'(0)$ exists at $\lim_{x \rightarrow 0} \frac{f(x)-f(kx)}{x} = l$

Let $f:(-a,a) \rightarrow \mathbb{R}$, with $a>0$. Assume $f(x)$ is continuous at 0 and such that the limit $$ \lim_{x \rightarrow 0} \frac{f(x)-f(kx)}{x} = l $$ exists, where $0<k<1$. ...
2
votes
0answers
34 views

Domain of log and its derivative

Let $f(x)=log(x)$ then the domain of function $f(x)$ will be $(0,\infty)$. My question is that what will be the domain of its derivative $f'(x)=1/x $? I think It should not be ...
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4answers
75 views

finding derivative of g(x) = xtanx

The question reads if $g(x) = x\tan x$, then the value of $g'\left(\frac\pi4\right)$ is: a) $1+ \frac\pi4$ b) $\frac\pi2-2$ c) $1-\frac\pi4$ d) $1+\frac\pi2$ I can never get to a situation where ...
0
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0answers
53 views

The definition of partial derivative

$$\lim_{\Delta z \to 0}\frac{\left(rN_{Az} \right) |_{z + \Delta z}-\left(rN_{Az} \right)|_z}{\Delta z}+\lim_{\Delta z \to 0}\frac{\left(rN_{Ar} \right) |_{r + \Delta r}-\left(rN_{Ar} ...
1
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1answer
36 views

Is this function symmetrical?

I have created a function $B_{n,k}(f'(x),f''(x),\cdots,f^{(n-k+1)}(x))_{(f \rightarrow g)^c}$ that behaves as follows: $$ B_{n,k}(f'(x),f''(x),\cdots,f^{(n-k+1)}(x))_{(f \rightarrow g)^c} = ...
1
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2answers
32 views

Find trig derivative of $y=4x(7x+\cot{7x})^6$

Find trig derivative of $y=4x(7x+\cot{7x})^6$. I got $y'= 4(7x+\cot{7x})^6 + 168x(7x+\cot{7x})^5 (\cot^2 {7x})$ but I'm not sure I did it right. Your help is appreciated (:
0
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1answer
20 views

Find the point $T(a,b)$ on the curve $y = x^2$ which has the shortest distance between itself and the point $P(3,0)$ [Solution Verification]]

Find the point $T(a,b)$ on the curve $y = x^2$ which has the shortest distance between itself and the point $P(3,0)$. $$ \\ \begin{align} \\ y &= f(x) = x^2 \\ b &= f(a) = a^2 \\ \\ &T(a, ...
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3answers
22 views

Derivate a logaritmic function

Let's take $ f = \ln(x) $. The derivate is $ f' = 1/x$. However $g = \ln(50x) $ has the same derivate $f' = g'$. How come? If I where going to derivate $g$ I would substitute $x$ for $t$: $g = ...
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3answers
21 views

Find the point $T(a, b)$ on the curve $y = x^2 + 1$ whose tangent passes through the point $P(1, 0)$ from the left.

Find the point $T(a, b)$ on the curve $y = x^2 + 1$ whose tangent passes through the point $P(1, 0)$ from the left. $$ \\ \begin{align} \\ \\ f(x) &= x^2 + 1 \\ f(a) &= a^2 + 1 = b \\ \\ T(a, ...
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5answers
158 views

Find the derivative of the function. y = $\sqrt{7x+\sqrt{7x+\sqrt{7x}}} $

This question is really tricky. I am wondering if I am right?
3
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1answer
67 views

Find the derivative of the function. $y = \cot^2(\sin θ)$

My work is as follows. Criticism welcomed. $$y = \cot^2(\sin\theta) = (\cot(\sin\theta))^2$$ Power Rule combined with the Chain Rule: $$\begin{align} y' & = 2(\cot(\sin \theta)) \cdot \frac ...
3
votes
2answers
66 views

Does the function $f(x)=x$, $x\in (0,1)$ have a maximum and minimum value?

My book says that since we cannot determine the value of x when it is just less than 1 and just greater than 0, hence the function does not have a maxima or minima. But the fact confuses me because ...
3
votes
1answer
51 views

Related Rates Question With Cylinder?

On a test we needed to solve the following question: A right circular cylinder with a constant volume is decreasing in height at a rate of 0.2 in/sec. At the moment that the height is 4 inches ...
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1answer
93 views

Formula for the $n^{th}$ derivative of $f(x)$

I am presented the following prompt: Find a formula for the $n^{th}$ derivative of $f(x) = \frac{x^n}{1-x}$ I've split the function into two parts to differentiate at the suggestion of some users (I ...
0
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1answer
36 views

logarithmic differentiation issue

Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. $$ 1/x(x+1)(x+2) $$ and I know that $$log((ab)/c) = log(a) + log(b) - log(c)$$ So I tried to ...
1
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0answers
20 views

In the space of polynomials of degree 2 or less, given the derivative linear transformation D and $T:=1+D+D^2$, $S:=1-D$, show that $S=T^{-1}$

Let $ P_2[X] $ be the space of polynomials of degree equal or less than 2 over the field R. Let: $$ D: P_2[X] \rightarrow P_2[X] $$ Be the derivative linear transformation, defined as follows: $$ ...
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0answers
39 views

proof of a series

Hi cant figure out how to prove the following: given $$ f(x)=(x-b)^5+(x-a)^4 $$ prove that the point $f'(c)=0$ divides the segment $[a,b]$ to a ratio of $4:5$ hint: use Rolle's theorm
0
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2answers
24 views

Example is required

I am trying to find a seuqence of a continuous functions $\{f_n\}$ defined on $[0,1]$ bounded by some small number, say $\varepsilon$ with the additional requirement of $f_n^\prime(t_0)=1$ at a ...
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1answer
26 views

Find the equation of the normal line to the function.

Here is the problem as well as my work: Am I correct? I am unsure if I correctly related the slope of the tangent line to that of the normal line..
2
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3answers
239 views

Derivative getting different result

Studying for a midterm, and one of the problems is: $$\frac{x^3+7}{x}$$ and we have to find the derivative. My professor is getting: $$2x-\frac{7}{x^2}$$ But I got $$3x-\frac{x^3+7}{x^2}$$ I even ...
4
votes
6answers
343 views

Find the derivative using the chain rule and the quotient rule

$$f(x) = \left(\frac{x}{x+1}\right)^4$$ Find $f'(x)$. Here is my work: $$f'(x) = \frac{4x^3\left(x+1\right)^4-4\left(x+1\right)^3x^4}{\left(x+1\right)^8}$$ $$f'(x) = ...
3
votes
1answer
52 views

surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
3
votes
3answers
98 views

Differentiate $\,y = 9x^2 \sin x \tan x:$ Did I Solve This Correctly?

I'm posting my initial work up to this point. Criticism welcomed! Using the formula $(fgh)' = f'gh+fg'h + fgh'$, differentiate$$y = 9x^2\sin x \tan x$$ $$\begin{align} y' &= 9\frac ...
0
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1answer
46 views

When finding the derivative of a function why do we have cancel out the x's in the numerator and the denominator?

I get why we cancel them out but I do not understand why we have to. Take $x^3+\frac{2}{x^2}$ for example. Why is $3x^2+\frac{2}{2x}$ wrong? Note: Please use terminology that someone just learning ...
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1answer
30 views

partial derivative in an exat equation

I have to determinate if this equation is an exact differential equation, but I don't now how get the partial derivative respect X & Y, I am confuse, please help me step by step I would ...
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1answer
33 views

How to calculate a Differentiable Quotient?

This is more than likely an Algebra problem but I can not figure out where the $-4x^2$ came from - first equation -3rd line. I do see that they transferred the $2\sqrt{x}$ to the denominator. What ...
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3answers
40 views

What is meant by “find the slope of the tangent to the graph of f at a general point x”

I am pretty thick and need questions to be specific or I do not know what they want. Do they want me to give a random example for x? eg the slope at x=7 is 5x?
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1answer
32 views

Derivative tests question

Show that $k(x) = \sin^{-1}(x)$ has $0$ inflections $2$ critical points $0$ max/min I find that the first derivative is $$\frac{1}{\sqrt{1-x^2}}$$ Second derivative is ...
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1answer
68 views

Definition of a 2-variable function derivative

I read this definition in a book of multivariable calculus: $f(x,y)$ is differentiable at $(x_0,y_0)$ if it can be expressed as the form $$f(x_0+\Delta x, y_0+\Delta y)=f(x_0,y_0)+A\Delta ...
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1answer
39 views

Differentiability of the function $x \mapsto |x|^{3/2}$ at $x = 0$

Could someone please explain whether the function $$\vert x \vert^{3/2}$$ is differentiable at zero? ($x$ here is a real number.) I tried investigating the right and left-sided limits (i.e., the ...
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1answer
22 views

The existence of the $n$th derivative at $c$ presumes the existence of the $(n-1)$st derivative in an interval containing $c$

The following is from Introduction to Real Analysis by Bartle. If the derivative $f'(x)$ of a function $f$ exists at every point $x$ in an interval $I$ containing a point $c$, then we can consider ...
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votes
3answers
62 views

Calculating arc length $y=x^2$

I picked this example for practice and got stuck with it. Someone moderate me if I am in the right path. I need to calculate the length of arc s, on the section of the curve $y=x^2$ with $0≤x≤1$ My ...
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votes
1answer
24 views

Differentiating hyperbolic functions.

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed. $\sech^{2} x$. $2\bigl(\cosh(2x) - ...
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1answer
32 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
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votes
2answers
66 views

Finding slope at a point in a direction on a 3d surface

This is not a duplicate, I have attempted the question and am not sure why my answer is incorrect. QUESTION: The surface with equation $z = x^3 +xy^2 $ intersects the plane with equation $ 2x−2y = 1$ ...
1
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1answer
42 views

Showing differentiability for a multivariable piecewise function

Let $$f(x,y)=\begin{cases} xy\sin(x/y) & y\neq 0 \\ 0 & y=0\end{cases},$$ show whether $f(x,y)$ is differentiable at $(0,0)$. It seems that there are multiple ways to do this but ...
0
votes
2answers
33 views

Find the absolute min and max in the given intervals

$k(x) = e^{-\frac{x^2}{2}}$ on $[-1,2]$ I think the derivative of that is $ -x e^{-\frac{x^2}{2}}$. I don't know how to find zero from that equation.
1
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1answer
56 views

Differentiation of the function $\operatorname{li}(x) = \int_2^x \frac{dt}{\ln(t)} $

I have to differentiate with respect to x: $$\operatorname{li}(x) = \int_2^x \frac{dt}{\ln(t)} $$ I havn't come across this before, so my idea is to integrate it first? (Backward right?). let ...