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

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2answers
53 views

Rate of change. Two ships sailing

Two ships. Ship A is 25 km south of ship B at 8 AM. If ship A is sailing west at 16 km/h and ship B is sailing south at 20 km/h, question asks for rate of change of distance between this two when it's ...
1
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1answer
39 views

How to take the derivative of a function $F(x)$

The function $F(x)=\int_{-1}^{x}\sqrt{1-t^2}dt$. I believe this to be the representation of the area under the curve between $-1$ and $x$, where $\int_{-1}^{x}\sqrt{1-t^2}dt$ is a function of $x$: ...
2
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1answer
40 views

Direction of gradient from level surface?

In the diagram below, we see a level surface with a gradient. As a consequence of the multivariable chain rule, the gradient is normal to the surface. That's clear to me. Why is the gradient ...
4
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4answers
360 views

Prove that $\int_a^c f(t)dt - \int_c^b f(t)dt = f(c)(a+b-2c) $, for some $c\in(a,b)$

Let $f$ be a continuous on $[a,b]$ then prove that there exist some $c$ that lies in $(a,b)$ such that $$\int_a^cf(t)\,dt - \int_c^b f(t)\,dt = f(c)(a+b-2c) $$ and hence prove that $\int_a^c ...
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2answers
39 views

How to prove the following? $\frac{d}{dx}a^x=(\ln a)a^x$

How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$ Just a hint will do it.
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1answer
49 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
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1answer
272 views

First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
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1answer
24 views

How to find the values of constants when there is one stationary point, no stationary point, and determining the maximum number os stationary points.

b) values of x is when f'(x) = 0 c) how do i solve this without using common sense and knowing that if a=0 there will be no turning points/inflections d)how do i solve this? e) max number of ...
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4answers
36 views

Determining local maximum or minimum - derivative worded problem

A cubic function has the rule $y=f(x).$ The graph of the derivative function $f'$ crosses the $x$-axis at $(2,0)$ and $(-3,0).$ The maximum value of the derivative function is $10$. The value of ...
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2answers
35 views

Re-writing a a differential function

I don't understand the concept of this... how do I derive a an equation written in terms of a function? How do I differentiate f(function inside) ...?
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0answers
33 views

Finding the constant of a function in terms of the gradient of a tangent.

Let $f : \Bbb R \to \Bbb R, f (x) = e^x+ k$, where $k$ is a real number. The tangent to the graph of $f$ at the point where $x = a$ passes through the point $(0, 0)$. Find the value of $k$ in terms of ...
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1answer
54 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
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1answer
54 views

Show that function is partially differentiable

I have the following function: $$F: \mathbb{R}^2 \rightarrow \mathbb{R}, ~~ (x,y) \rightarrow xy\frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \ne 0$ and $F(0,0) = 0$. I want to show that $F$ is partially ...
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2answers
52 views

Calculus 1 Proof

How do I prove the following statement? Suppose that $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ and $$\lim_{x\to a^+}f'(x)=L.$$ Show that the right hand derivative at $a$ (consider ...
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1answer
40 views

get the length of a curve with integral

I need to get the length of a curve which equation is : $$y= (4-x^\frac{2}{3})^\frac{3}{2}$$ I need to find the length using the method : $$L=\int_a^b \sqrt{ 1 + \left(\frac{dy}{dx}\right)^2}$$ So ...
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1answer
39 views

Algorithm to compute Newton polynomial derivative

I'm unable to find a clean solution to this problem and hope someone here can help me. Given a list of x-values: $x_0, x_1, x_2, ... x_n$ and a value $x$, I want to determine the accompanying parts of ...
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0answers
13 views

Derivative in given direction

Trying to find the derivative in one direction I got stuck a bit. In particular let's say I have a pair of orthogonal axis $(i,j)$ such as they form an angle of $\theta$ with $(x,y)$. In particular ...
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2answers
178 views

Do you feel comfortable with integral u-substitution? (reverse chain rule)

I've made this post both to see if I'm thinking right and to let others read and understand where the "u-substitution" method for integration comes from. I really hate substitutions, because you lost ...
4
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1answer
47 views

Speediness and correctness when graphing by hand .

First of all thank you for visiting this question! I believe it's a pretty simple problem but get's kinda hairy and time consuming on each step as I have done it, so my question (the one you are here ...
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0answers
13 views

Differentiability of a composition: minimal regularity assumptions

Let $f \colon \mathbb R^n \to \mathbb R$ be a $C^1$ function and let $\phi \colon \mathbb R \to \mathbb R^n$ be a $C^1$ curve. Then it is well known that the composition $f \circ \phi$ is ...
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1answer
92 views

Function which derivative at $0$ is $1$ but is not monotonic increasing

Please, I need help in order to understand the following assumption that I've found in Bartle's book Introduction to Real Analysis page 171. It says: One might suppose that,if the derivative is ...
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0answers
40 views

Second Derivative of log

Let: $\log(s)=z$ I understand that $$\frac{\partial}{\partial s}=\frac{\partial}{\partial z} \frac{\partial z}{\partial s} = e^{-z}\frac{\partial}{\partial z}$$ What is the second derivative, ie ...
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0answers
23 views

Characterization of the derivative as a tensor field

I was thinking about the derivative, and I wanted to make sure I’m thinking about it the right way. Suppose we have a $C^{\infty}$ function $f: {V}\to \mathbb{R}$, where $V$ is a finite-dimensional ...
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1answer
31 views

Finding the point where a function turns smaller then another

Sorry, couldn't explain better on the title. I mean, if you have a function for the income over time $I(t)$ and another one for costs $C(t)$ and you want to find out the time $t$ for which the profit ...
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1answer
34 views

Differential Calculus Problem - Sphere volume increasing (differentiation of algebraic functions)

The Air is pumped into a spherical ball which expands at a rate of 8cm^3 per second. Find the exact rate of increase of the radius of the ball when the radius is 2 cm. I have tried this question, ...
3
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2answers
44 views

derivative after changing variable

I have just studied a lesson about derivative of a function but I still confuse in the following case. Suppose that I have a function: $$ f(x) = 2x^2 + 3x + 1$$ and I want to calculate ...
2
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1answer
42 views

Explain the minus sign in the following formula.

I just read that: If $z=f(x,y)=c$, be the equation of a curve, then the slope of the tangent to the curve at any point (x,y), is given by $$m=\frac {dy}{dx}=-\frac{\frac{\partial z}{\partial ...
2
votes
4answers
129 views

When can I say that $f(x) \gt g(x) \implies f'(x) \gt g'(x)$?

Are there cases when this relation holds? $$f(x) \gt g(x) \implies f'(x) \gt g'(x)$$ I.e. what are the conditions on $f(x)$ and $g(x)$ for that to be true? Is it even possible to determine them? In ...
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1answer
40 views

If a differentiable function has bounded derivative, Must it be that its derivative continuous?

I got this question: Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is ...
4
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1answer
61 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
4
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1answer
39 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
3
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4answers
99 views

Lines tangent to parabola at point.

I'm struggling to figure out what I'm exactly required to do. The problem states "Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$." I believe ...
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2answers
52 views

What does d f(t,x) = 0 mean?

A differential equation that can be written in the form $d\phi(t, x) = 0$ for some continuous and differentiable function $\phi(t, x)$ is called exact. What does $d\phi(t, x) = 0$ mean?
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2answers
37 views

Second derivative of $\frac{\ln t}{\sqrt t}$ and derivative of $\arccos(1-2x^2)$

$f(t)=\dfrac{\ln t}{\sqrt t}$ I'm stuck on the algebra of finding the second derivative. For the first derivative, I got: $f'(t)=\dfrac{t^{\frac{-1}{2}}(1-\frac{1}{2}\ln t)}{t^2}$ For the second ...
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2answers
62 views

Proof that energy of a free body is constant, using the derivate

Ok, what I'm trying to prove is the law of conservation of energy for a free fall. Let the downward direction be positive. We want to prove that: $$mgh+\frac{mv^2}{2}=constant$$ For this, we try to ...
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1answer
38 views

prove that a function whose derivative is bounded also bounded

I got this problem: Let $f$ be a differentiable function on an open interval $(a,b)$ such that $f'$ (the derivative of $f$) is bounded on $(a,b)$ (meaning there exist $0<M$ such that $\forall ...
2
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0answers
25 views

Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
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2answers
59 views

Finding $\frac{d}{dx} y^x$

$$\frac{d}{dx} y^x$$ How would you find the derivative with respect to $x$ of $y^x$ assuming that $y$ is a function of $x$? I know you will have to use the chain rule somehow, and I know that the ...
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1answer
18 views

Derivative of rigid motion like reflection?

Is it possible to define a derivative for rigid transformations eg. reflection and translation? I am especially interested on reflections shortly $\sigma$. Because I am trying to relate ...
5
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1answer
182 views

Pointwise boundedness is uniform for a sequence of derivatives

Let $f \in C^\infty ([a, b])$ be an infinitely differentiable function defined on a closed interval $[a, b]$ with the following property: for any $x \in [a, b]$ the sequence $|f^{(n)}(x)|$ is bounded, ...
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0answers
36 views

Maximizing sum with a constraint

Given the function $$ f(\alpha_{1},\ldots,\alpha_{k})=C\sum_{i=1}^k \alpha_i e^{-(b^2/d)\alpha_i}\text{ with } C>0,\ b>0,\ d>0,\ \forall i\in\{1,\ldots,k\}:\alpha_{i}\ge 0 $$ with the ...
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1answer
275 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
0
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1answer
42 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
2
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1answer
84 views

How to derive $\frac{d}{dx}\left(x+1\right)^{\sin\left(x\right)}$

I need help to find derivative of: $\frac{d}{dx}(x+1)^{\sin x}$ i tried to do something like this.. $$(x+1)^{\sin x}\cdot \ln\left(x+1\right)=\sin x(x+1)^{\sin\left(x\right)-1}\cdot ...
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1answer
89 views

Why continuity at a point one of Dini derivatives implies the continuity at this point others Dini derivatives?

Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function and $x_0\in (a,b)$. How to prove that if the Dini derivatives $D^+f(x_0)$ is finite and continuous at $x_0$ then also $D_+f(x_0)$ is ...
5
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1answer
138 views

How to show that $e^x$ is differentiable?

I tried to search for a few minutes but I didn't find this question so I hope it's not a duplicate. So I want to show that $(e^x)' = e^x$. To do that, I must proof that the limit: ...
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5answers
266 views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
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0answers
33 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
9
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1answer
259 views

Find compressed form for cumbersome calculation

Given the three functions $u^{\mathrm{(I)}}(t)\;=t \left(t^2\right)^{k}\,e^{2\beta t^2},\\ u^{\mathrm{(II)}}(t)=\sqrt{\left(t^2\right)^{2k}-\left(t^2\right)^{2k+1}}\,e^{2\beta t^2},\\ ...
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0answers
19 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...