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

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9 views

What is the Hessian matrix of $x\mapsto f(Ax+b)$?

Let $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^n$ $f\in C^2(\mathbb{R}^n)$ and $\tilde{f}(x):=f(Ax+b)$ for $x\in\mathbb{R}^n$ It's easy to prove that $$\nabla\tilde{f}(x)=A^T\nabla f(x)$$ ...
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0answers
23 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
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0answers
12 views

Does $f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$ have a local minimum or maximum at $-5$ or $5$?

Does $$f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$$ have a local minimum or maximum at $-5$ or $5$? I have discovered using the second derivative test that it has a ...
2
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1answer
21 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
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0answers
9 views

Curve torsion through $\mathbf{r}$

While learning torsion i came across formula $$\tau = \frac{\mathbf{r}'\mathbf{r}''\mathbf{r}'''} {\mathbf{r}''\cdot\mathbf{r}''} = ...
1
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2answers
24 views

How to find $ \frac{d (\tanh(kx))}{d x}=?$

I am tried to resolve the problem $$ \frac{d (\tanh(kx))}{d x}=?$$ where $k$ is positive value. I found one solution that is $$ \frac{d (\tanh(kx))}{d x}=\frac{k}{2\cosh^2(kx)}$$ Is it right? If ...
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1answer
27 views

How to find dy/dx = - fx/fy?

I need some walkthrough in solving the following question: find dy/dx = - fx/fy? 3x^2 - y^2 + x^3 = 0. I need to know the method to solve this question. ...
1
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0answers
22 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
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0answers
23 views

Find the derivative of average rate of change in $[a, x]$

Let $g(x)$ be average rate of change of $f(x)$ in $[a, x]$ Find the $g'(x)$ where $x \in \mathbb{R}$ satisfy $x > a$. I don't understand what the problem says.
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0answers
24 views

how to solve this equation to put “h” as function of time? [on hold]

I'm trying to get an equation to define the height (h) as function of time. How can I solve this?
9
votes
3answers
206 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
2
votes
1answer
40 views

TI-84 gives 100 for d/dx(cube_root(x)) at x=0

My TI-84 Silver Edition is doing something strange. If $f(x)=\sqrt[3]{x}$, $\frac{d}{dx}\sqrt[3]{x}=\frac{1}{3\sqrt[3]{{x^2}}}$ At $x=0$, $\frac{d}{dx}f(0)$ is undefined. When I type ...
1
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1answer
17 views

Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - ...
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1answer
20 views

Find the matrix A of “T” in the basis B [on hold]

Let V ={asin(x) + bcos(x)| a,b element of R} with ordered basis B = (sin(x), cos(x)). Let @: V->V be defined by @(V)= d(v)/dx (differentiation). Find the matrix A of @ in the basis B. Please help. ...
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2answers
34 views

If $f(x)=\sin^2(3-x)$, then what is $f'(0)?$

I've been doing the math myself and my answer happened to be $-\sin(6)$, am I just being really stupid here and unable to convert it to any of the answers or my answer is wrong (or the answers are ...
1
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2answers
32 views

Is it true that $\frac{d}{dt}f(g(t),h(t))=f'(g(t),h(t))g'(t)+f'(g(t),h(t))h'(t)$

I want to solve the following question: We want to find $\frac{du}{dt}$ where $u(x,y)=x^2y^3$ and $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$. I know we can just plug $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$ in ...
2
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1answer
24 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
3
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2answers
91 views

If $\lim_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim _ {x\rightarrow \infty} f(x) = 0$ [duplicate]

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function and $\lim_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim _ {x\rightarrow \infty} f(x) = 0$. I really have ...
3
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2answers
30 views

Find derivative of integrate square function [on hold]

I am finding a solution of that function. Could you have me to resolve it $$F=\left( \int {(ax+b-c)}^2 dx \right) +\lambda_1(a-m)^2+\lambda_2(b-n)^2$$ where $c,m,n ,\lambda_1,\lambda_2$ are constant ...
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0answers
27 views

Derivative of inv: subset of linear automorphisms

I have no clue how to approach this problem, I've asked for some help from different people, but I have yet to comprehend it. The question is the following, Let $\mathcal L$($\mathbb C$$^n$) denote ...
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3answers
45 views

Power rule vs. Derivative rule

I have been learning about derivatives and need some answers. So the power rule is simple you just bring down a power such as $f(x)=x^2$ becomes $f'(x)=2x$. Then with the derivative rule we use the ...
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1answer
52 views

Find the derivative of $\arcsin(x)$ by just using the common rules

I need to find the derivative of $\arcsin(x)$ by just using the common rules of differentiation, such as sum, scalar multiplication, product, quotient rule, the chain rule and the inverse function. ...
2
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1answer
20 views

find the derivative of a function with natural log

find the derivative of $f(x)=\ln(x^4)(\sqrt{5x-3})$ I just need help getting to the answer. The first answer I got was $f(x)=(x^4)(2.5)+(5x-3)^{1/2}(4x^3)$.
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1answer
19 views

Derivation with TI-nspire CX CAS not working

I try to get the partial derivative of: $$\frac{x^2y-xy^2}{2}$$ where $x \in \mathbb{R}$ with respect to $y$.. When I type it into my TI-nspire I get $xy$, however the answer should be ...
0
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1answer
26 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
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1answer
24 views

derivative calculation is more efficient than the integration calculation

I read that for $n$ sampled time points, the computation time required by the derivative calculation increases linearly with $n$, while the computation time required by the integral calculation is ...
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1answer
29 views

Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$?

I am struggling with derivatives of exponents functions... Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$? Where does ...
0
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3answers
33 views

Why derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$

I simply cannot understand why the derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$? Also, is that $\log$ the natural log or what?
0
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1answer
30 views

Find the number of divisors of $f'(1)$

The question is that: Let $f(x) = x^{25} + 2x^{24} + 3x^{23} + 4x^{22} + \cdots + 25x$. Find the number of positive divisors of $f'(1)$. How to find this number easily? Is there only one way: ...
0
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1answer
44 views

Calculus Derivative I have the work for the problem I'm just not sure where one part comes from

I'm new to this so I don't know how the formatting works sorry. So I have all the work for it there is just one thing I don't understand where it is coming from.... I know its a double chain rule ...
1
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1answer
32 views

Physical significance and graphical point of view of second derivative of a function $f''(x)$ .

This was just going through my mind - $f'(x)$ represents slope of a function. Then what does $f''(x)$ represent? For example, we define strictly increasing and strictly decreasing functions in some ...
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2answers
34 views

Derivatives of Sets

So, I often see this: $$D_x \left\lbrack \int f(x) \right\rbrack = f(x)$$ But this is a derivative of a set of antiderivatives. What is the conceptual backing for this i.e. what does it mean to derive ...
4
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4answers
320 views

Are derivatives linear maps?

I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this $$\lim_{h\rightarrow 0} ...
3
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3answers
66 views

Why the derivative of $n^{1/n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$

Why the derivative of $n^{1/n} = \sqrt[n]{n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$ (according to Maxima and other tools online)? I have tried to applied the chain rule, but ...
3
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0answers
72 views

40th derivative of a function

I would like to have some verification to see if my answer is correct. The given function is $f(x)=ln(1+x^2)$ and I need the 40th derivative at $x=0$. Here is my work: Using series one can manipulate ...
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0answers
6 views

Backwards finite difference for mixed partials at higher order

I'm trying to understand what is the general method for calculating a backwards difference for a mixed partial of $n$ variables. Let's start with one variable: The forward and backward finite ...
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0answers
17 views

What is the difference between an integral and an antiderivative and how do their respective relations to the FTC differ?

I'm currently taking Calculus and I'm having difficulty understanding the difference between integrals and antiderivatives. Yesterday when the concept was introduced in lecture, I was under the ...
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2answers
27 views

On a connection between Newton's binomial theorem and general Leibniz rule using a new method.

In calculus the general Leibniz rule asserts that Let $n$ be a natural numbers, if $f$ and $g$ are $n$-times differentiable functions at a point $x$, then the function $fg$ is also $n$-times ...
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6answers
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What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
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2answers
34 views

Partial derivitive of a summation.

I need some help taking the partial derivative of this function, if it is possible. Thanks!
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1answer
33 views

Solve $3x³ + 3y³ + 2x² - 32 = 0$, $4x² + 2 = 0$ and $10y² + 2x² + 12 = 12x³$.

Hi my friend asked this to me, i'm not good at math. $$3x³ + 3y³ + 2x² - 32 = 0$$ $$4x² + 2 = 0$$ $$10y² + 2x² + 12 = 12x³$$ remove 2x² $$2x² = -1$$ $$3x³ + 3y³ - 1 - 32 = 0$$ $$10y² - 1 + 12 = ...
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0answers
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Finding the marked values of x on f(x) and its derivatives.

I posted this question yesterday but I didn't get a proper answer on how to find the values of f(x) and its derivatives and the greatest and least values. I understand how you can find the values of ...
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0answers
12 views

Function whose fixed points are a convergent sequence with derivatives at each term $\neq 1$ and derivative at limit point $=1$

The question asks to find an example of a function where the fixed points of the function are a sequence that converges to a fixed point, where the derivative of the function at each of the fixed ...
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2answers
65 views

”lesser known” rules to calculate the derivative

I was reading through the online help of WolframAlpha (link) and found this statement: Wolfram|Alpha calls Mathematica's $D$ function, which uses a table of identities much larger than one would ...
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2answers
30 views

Trying to solve for x when you have sine and cosine in the function

Graph $f(x) = \sin (x^2)$ I need to graph $f(x) = \sin (x^2)$ from $-2\pi$ to $2\pi$ and from that, I need to include the first derivative which is set to zero and used to find the maximum and ...
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2answers
26 views

Derivative of a Trigonometric Function with Cosine and Sine to find the Maximum

The movement of the crest of a wave is modelled with the equation $h(t) = 0.2\cos(4t) + 0.3\sin(5t)$. Find the maximum height of the wave and the time at which it occurs. I have no idea how to go ...
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0answers
15 views

show F is differentiable at a point P in $\mathbb R^2$

I am trying to use the following definition to show that F is differentiable, $F(a+h,b+k)-F(a,b)=L(h,k)+\epsilon(h,k)$ where L is the linear part and I hope epsilon is small. Firstly, I am not ...
0
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2answers
34 views

Derivatives Applications Problem

My teacher dropped this problem in class but no one seemed to answer it correctly. Any help? A man is walking from his front door to his car. The length of garden from his door to the road is 4 m. The ...
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1answer
17 views

There exists a continuously differentiable bijection, $g:[a,b]\to [c,d]$ satisfying $g'(k)>0$ with $z(k)=w(g(k))$

Let $z:[a,b]\to \mathbb{C}$ and $w:[c,d]\to \mathbb{C}$ such that there exists $t(s):[c,d]\to [a,b]$ which is a continuously differentiable bijection with $t'(s)>0$ and $w(s)=z(t(s))$. Then I ...
0
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1answer
37 views

Use calculus to derive area of circle using n triangles

This is a homework question I am struggling with... Let $n$ be a positive integer, and cut the circle into $n$ equal sectors. In each sector there is an isosceles triangle formed where the edges of ...