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

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0
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1answer
27 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 ...
0
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2answers
61 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 ...
84
votes
6answers
10k views

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
43 views

Partial derivitive of a summation.

I need some help taking the partial derivative of this function, if it is possible. Thanks!
-1
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1answer
46 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 = ...
0
votes
0answers
14 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 ...
4
votes
2answers
75 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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vote
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 ...
0
votes
2answers
32 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 ...
0
votes
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 ...
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2answers
105 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 ...
1
vote
1answer
18 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
votes
1answer
75 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 ...
3
votes
1answer
32 views

Show that Vandermonde-like $3 \times 3$ determinant is non-zero

I want to show that the determinant $$ f(t)=\det \begin{bmatrix} 1 & x & x^t \\ 1 & y & y^t \\ 1 & z & z^t ...
0
votes
1answer
16 views

Hessian determinants and meanings

Suppose $f : \mathbb{R}^2 \mapsto \mathbb{R}$ is a $C^2$ map. Can the determinant of the Hessian matrix of $f$ at the same point be different for different bases of $\mathbb{R}^2$ ? What about eigen ...
4
votes
0answers
39 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then ...
3
votes
1answer
12 views

Derivative limit is uniformly convergent

If we consider the sequence of functions: $g_{n}(x)=\frac{f(x+h_{n})-f(x)}{h_{n}}$ where $h_{n}>0$ is a sequence of real numbers converging to $0$, and $f$ is a $C^{1}$ function. How can you show ...
0
votes
1answer
28 views

Separating variables in the PDE $u_{tt}+2u_t+u=u_{xx}$

Separating variables in the PDE $u_{tt}+2u_t+u=u_{xx}$ In the ODE of $T(t)$ (Second last equation) shouldn't it be the one stay outside the bracket?
1
vote
1answer
38 views

L'Hospital rule kind of

Let $f,g:[a,b]\to\mathbb{R}$, $a,b\in\mathbb{R}, a<b$ be two smooth functions (i.e. $f,g\in C^{\infty}([a,b])$. Is it true that, if for a fixed $t_0\in (a,b)$ with the property that$\ g(t)\neq ...
0
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1answer
34 views

Almost everywhere differentiability

Suppose $f: \mathbb{R} \to \mathbb{R}$ is increasing and $g = f$ almost everywhere with respect to Lebesgue measure (a.e.). Suppose $g'$ exists a.e. Does it follow that $g' = f'$ a.e.? This comes ...
1
vote
1answer
19 views

Chain rule version for partiel derivative?

Non-math student here so go easy on me. How do we calculate a partial derivative in terms of $x$ when dealing with a multivariable composite function, and what 'chain rule version', if any, could one ...
4
votes
2answers
94 views

What's so special about radians? (Differentiation) [duplicate]

It seems to me that radians have lots of very special properties that allow us to do maths with trigonometric functions. When I first came across radians, I was led to believe that they were designed ...
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votes
3answers
24 views

Determine the area of ​​the shape bounded by the curves

$$y=x^2 + 1, x=2, x=1, y=0$$ I've got exam today and I'm learning how to solve this type of task. The exam is about derivations mostly.
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0answers
8 views

Schwartzian derivative of a vector function

I am not a mathematician and hence I really do not know where to look for answer. I have a system which is governed by the three equations. say $x_{n+1} = f(x_n,y_n,z_n)$ ,$y_{n+1} = ...
0
votes
2answers
47 views

Using implicit differentiation with a fraction

How do I solve this? What steps? I have been beating my head into the wall all evening. $$ x^2 + y^2 = \frac{x}{y} + 4 $$
2
votes
1answer
29 views

Finding the marked values of x on a graph

I would assume that since $x_3$ is the local maximum(or absolute maximum) on the graph of $f$ prime, that it would be the greatest on the graph of $f.$ However, this problem is online, and in ...
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2answers
48 views

Find $f'(2)$, where $f(x) =\frac{h(x)}{x}$.

Consider the function $h(x)$, for which $h(2) = 4$ and $h'(2) =-3$. Find $f'(2)$ for the function $f(x) = \frac{h(x)}{x}$. Progress: I know that $h(x)/x$ is equivalent to $h(x) x^{-1}$; should I ...
0
votes
0answers
27 views

How fast is this dot moving when the angle $θ$ between the beam and the line through the searchlight perpendicular to the wall is $π/6$?

A searchlight rotates at a rate of $4$ revolutions per minute. The beam hits a wall located $11$ miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per ...
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votes
3answers
46 views

How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t} $$ $$y=\sin(t) \cdot e^{-t} $$ Show that $dy/dx =tan(t-\pi/4) $. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
0
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0answers
18 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
1
vote
1answer
52 views

Prove $f'''(x) \geq 3$ for some $x \in (-2,2)$, if $f$ is cont on $[-2,2]$ and three times differentiable in $(-2,2)$ & $f(2)=-f(-2)=4$ & $f'(0)=0$

Prove that there exists a $x \in (-2,2)$ such that $f'''(x) \geq 3$, if $f$ is cont on $[-2,2]$ and three times differtiable in $(-2,2)$ with values $f(2)=-f(-2)=4$ & $f'(0)=0$. How do I handle ...
0
votes
0answers
36 views

Increasing/ decreasing functions

We are given a random variable x with a pdf f(x) and F(x) is its distribution function. Let $$r(x) = \frac {xf(x)} {1-F(x)} $$ Then for $x< e^{\mu} $ and $$f(x) = \frac {e^ {1/2(\log x - \mu)^2}} ...
0
votes
1answer
30 views

Derivation of Ln root

$$f(x) = \ln \sqrt{4x-3}$$ I'm practicing derivation for the exam and I'm stuck on this task. Could someone help me out in solving this. But when it comes to root, I'm a bit confused. The result is ...
1
vote
1answer
47 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
1
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2answers
25 views

Let $f(x) = |x − 1| + |x − 2|^3$ for $x\in \mathbb{R}$ . Then examine the differentiability of $f$ at x = 1 and x = 2 and rest of $x\in \mathbb{R}$ .

Let $f(x) = |x − 1| + |x − 2|^3$ for $x\in \mathbb{R}$ . Then examine the differentiability of $f$ at x = 1 and x = 2 and rest of $x\in \mathbb{R}$ . derivative of the function f at a: ...
0
votes
1answer
33 views

directional derivative problem

for a point M(4,1) and a function $z = x y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the $x$ axis....I find it a little ...
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vote
2answers
47 views

Jacobi's Derivative of the Determinant

I've been given the following theorem for the derivative of the determinant of a matrix: "Let $A\in \mathbb{R}^{n\times n}$ be a square matrix. Then the Fréchet derivative of det$: ...
2
votes
1answer
52 views

Diffeomorphism from disk to plane

I want to show that the disk $D = \{(x,y) \in \mathbb R^2 : x^2+y^2 < 1\}$, the open square $K = (-1, 1)^2$ and the whole plane $\mathbb R^2$ are all diffeomorphic to each other. Therefore I want ...
2
votes
5answers
153 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
0
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1answer
30 views

Quotient rule for derivatives..am I making this to complicated

This is a straight forward question.. When I have something like 10/x (i.e basically whenever the numerator is just a number with no variables) and I need to take the derivative I go through the ...
0
votes
1answer
45 views

Using Taylor series find derivatives of arctan(x)

Using Taylor series for $arctan(x)$, find $f^{(5)}(0)$ and $f^{(6)}(0)$ for $f(x)=arctan(x)$ I figure for this problem I compare the general Taylor series formula to the Taylor series for $arctan(x)$ ...
2
votes
0answers
50 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
1
vote
1answer
57 views

If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
1
vote
1answer
29 views

Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
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vote
2answers
26 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.
3
votes
2answers
33 views

Differentiating inverse hyperbolic function

I am trying to differentiate $\tanh^{−1}\left(x/(1 + x^2)\right)$, but am finding it difficult understanding what to do. I think you have to place the differential of the angle of the hyperbolic ...
1
vote
3answers
30 views

Differentiating inverse trig function

When differentiating $\sin^{-1}(x/2)$, I got $\frac{1}{2}(4-x^2)^{-1/2}$ but the answer I'm given does not include being multiplied by half. Can anyone explain if the answer I'm given is right and ...
1
vote
2answers
49 views

Differentiation of a constant function from first principles

How do you differentiate a constant $K$ from first principles to show that it equals zero? $f(x) = K$ but what does $f(x+h)$ equal to where $h$ is the change in $x$?
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2answers
41 views

Stationary points of $ \ln(x + 1)$?

I'm trying to find the stationary points of $f(x) =\ln(x + 1)$. When I differentiate, I get $f'(x) =\dfrac{1}{ (x + 1)}.$ I then set that to zero and end up getting $1 = 0$? I'm not sure what this ...
4
votes
2answers
60 views

Find the solution to the differential equation

Assume $x\gt 0$ and let $$x(x+1)\frac{du}{dx} = u^2,$$ $$u(1) = 4.$$ I started off by doing some algebra to get: $$\frac{1}{u^2}du = \frac{1}{x^2+x}dx.$$ I then took the partial fraction of the ...