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

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2
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2answers
59 views

Finding the absolute minimum and maximum of a function

The function is $$f(x)=x+\sin(2x)$$ I need to find the absolute maxima and minima of several different domains using this function. I have found that the derivative of this function is ...
1
vote
2answers
30 views

A question on a multivariable continuously differentiable function

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that $x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect ...
1
vote
1answer
51 views

Computing the derivative of an integral

There are similar questions on the same topic, yet I could not figure out why the following equation (taken from an economics solution manual) holds: $$ \frac{\partial}{\partial C(i,j)} ...
0
votes
1answer
91 views

Is it true that for Increasing function $f'(x) \ge 0$?

In Spivak he claims that, "it's easy to see that for an increasing function $f'(x) \ge 0$". How? Increasing function is defined as: if $x \lt y$, then $f(x) \lt f(y)$. We are also assuming that ...
2
votes
1answer
35 views

Can any function on naturals be interpolated to a smooth function on reals?

Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that for any $n ...
1
vote
1answer
62 views

how to find slope of this polar curve: $r^2=\sin(2\theta)$.

Given $r^2=\sin(2\theta),\;$ how to find the slope of the tangent line at $x=0$ ? If the question were $r=\sin(2\theta)$, it would be o.k. but since it is $r^2=\sin(2\theta)$, I don't know how to ...
1
vote
1answer
19 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
0
votes
1answer
32 views

derivate of indicator function

What is the derivative of the indicator function: \begin{equation} f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases} \end{equation} thank you
3
votes
2answers
63 views

Show there are $b, c \in \mathbb{R}$ such that $f(x)= {a\over 2}x^2 + bx + c$

Given $f:I\rightarrow \mathbb{R}$ and $f''(x) = a.\forall x\in I$. Show there are $b, c \in \mathbb{R}$ such that $f(x)= {a\over 2}x^2 + bx + c$. If we define $g(x) = ax + b$, then $g'(x) = ...
0
votes
1answer
38 views

What is the derivative of a skew symmetric matrix?

I'm trying to work out some Jacobians and I ran across a problem. If I have a function of a vector making it a skew symmetric matrix, like below, what is the derivative $f'$? $$ ...
1
vote
1answer
25 views

Differentiability of “positive part” function

Let the positive part function be defined as $\max(0,x)$; this function is obviously not differentiable in $x=0$. But what about the (more smooth) function $\big( \max(0,x) \big)^{2}$. I suspect the ...
1
vote
2answers
58 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
0
votes
2answers
23 views

How do I get to the answer that is given using normal differentiation and standard rules of exponents?

Given that $y = \dfrac{x^3 - 5x}{\sqrt{x}}$, show that $\dfrac{dy}{dx}$= $\dfrac{5(x^2 - 1)} {2 \sqrt{x}} $ using standard rules of exponents. I get as far as $\dfrac{dy}{dx}= \dfrac {5}{2}x^\frac ...
3
votes
2answers
48 views

How do I differentiate to what they have given?

Given that $y = \dfrac{x^3 - 5x}{\sqrt{x}}$, show that $\dfrac{dy}{dx}$= $\dfrac{5(x^2 - 1)}{2 \sqrt{x}}$ (Posted from ``answer'' below: I get as far as $\dfrac{dy}{dx}= \dfrac {5}{2}x^\frac ...
6
votes
3answers
162 views

optimal way to approximate second derivative

Suppose there is a function $f: \mathbb R\to \mathbb R$ and that we only know $f(0),f(h),f'(h),f(2h)$ for some $h>0$. and we can't know the value of $f$ with $100$% accuracy at any other point. ...
0
votes
2answers
50 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
vote
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
votes
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
votes
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 ...
0
votes
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.
2
votes
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?
0
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1answer
259 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 $n$?
1
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1answer
23 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 ...
0
votes
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 ...
0
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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) ...?
0
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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 ...
0
votes
1answer
52 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 ...
0
votes
1answer
53 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 ...
0
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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 ...
1
vote
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 ...
1
vote
1answer
38 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 ...
0
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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 ...
6
votes
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
votes
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 ...
0
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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 ...
1
vote
1answer
89 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 ...
1
vote
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 ...
1
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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 ...
1
vote
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 ...
0
votes
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
votes
2answers
42 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
votes
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 ...
0
votes
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
votes
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
votes
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
votes
4answers
96 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 ...
1
vote
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?
2
votes
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 ...
0
votes
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 ...