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
34 views

Derivative of a definite integral (FTIC)

$\displaystyle \left.\frac{d}{dx}\right|_{x=\pi} \int_{t=0}^{x} \frac{\cos 3t}{\sqrt{1+t}} \,dt$ What would be the correct way to solve this? I'd really like to understand the reasoning behind it as ...
1
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2answers
40 views

Derivatives of Logarithmic functions

I am stuck in these problems. $\displaystyle \frac{d}{dx} (\log_2 x^8)$ $\displaystyle \frac{d}{dx} (e^x \ln x)$ I think for the first problem the answer is $\dfrac{2}{x^7}$, whereas for the ...
1
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1answer
37 views

How do I setup the lagrangian for this problem?

I have a function $y(x)$, that I would like to maximize, subject to two constraints. It is given by: $$ \max_{x} \ y(x) = a \ cos(x) + b \ sin(x) \\ \text{subject to:} \\ x \geq 0 \\ x \leq ...
2
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0answers
41 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
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1answer
58 views

Minimize a trig function. Getting stuck.

So I have just about given up on this. Here is the problem. FYI, all angles are in degrees, and $L$, $R$ are just strictly positive scalars. I have a trig-function $D$. Its derivative shown below, ...
1
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1answer
54 views

Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
4
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1answer
70 views

Is this Frechet derivative correct?

Problem statement: Let $u \in L^2[0, 1]$ and $$J(u) = \int_0^1 u(t) u(1-t)dt$$ Find $J'(u)$ and $J''(u)$. Attempted solution: First derivative There is a hint that the derivative looks like this: ...
0
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1answer
20 views

Derivative problem with give values

Hello please help me solve this, I really thought my asters for the last two parts were correct but apparently they were not. for part f i did 4 * -4 because that is what x * f'(x) would be equal to.. ...
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4answers
47 views

Chain Rule Problem

Suppose we wanted to differentiate the function $$h(x) = (2-2x^3)^4 + \frac{1}{2-2x^3}$$ using the chain rule, writing the function as the composite $h(x) = f(g(x))$. Identify the functions $f(x)$ ...
2
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1answer
72 views

Differentiability in $R^n$

I have the definition of the derivative for $f:\mathbb R^n \rightarrow\mathbb R^m$ at a point $a$ as: $f$ is differentiable at a then there exists a linear map $L:\mathbb R^n \rightarrow\mathbb ...
0
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1answer
65 views

Differentiation of the transpose of a vector? [closed]

Suppose $s$ is a scalar, and $x$ is a vector, how would I calculate $$ \left(\frac \delta {\delta x} (x^T s)\right) $$Basically I couldn't find any reliable source letting me know how to ...
1
vote
1answer
104 views

Show that the directional derivative is linear by definition

If $f$ is differentiable at $x$, the map $h\mapsto f(x+h)-f(x)$ should be approximately linear. The scalar multiplicativity can be seen by noting that $$\lim_{h\to 0}\frac{f(x+ch)-f(x)}{h} = ...
0
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1answer
62 views

How can I solve this indefinite integral?

Can someone please show me with steps on how to evaluate this indefinite integral?
2
votes
2answers
54 views

How can I solve this definite integral ?

How can I solve this? I need help with the steps
2
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2answers
161 views

Solve the following differential equation: $xy' - y = x^2$

I'm preparing to exam in Linear Algebra $2$ and I have problems with differential equations.. For example, the following exercise: Solve the following differential equation: $xy' - y = x^2$. I ...
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0answers
35 views

Question concerning the Lie derivative and the Lie bracket

Let $X,Y$ be vector fields on a differentiable manifold. In a proof I read that for a special chart (namely the chart in which we have $X\equiv e_1$) it holds ...
1
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2answers
79 views

Increasing function on small interval given positive derivative

Suppose that $f'(0)>0$. Does it imply that there exists a $\delta > 0 $ such that $f$ is increasing on $[0,\delta]$? I think this is false and I've been trying to think of a counter example. I ...
0
votes
3answers
90 views

Calculus Derivatives - Finding a function, given tangent and x intercepts

I am having trouble solving this problem: Find numbers $a$, $b$, and $c$ so that the graph of $f(x) = ax^2 + bx + c$ has $x$-intercepts at $(0,0)$ and $(8,0)$ and a tangent with slope $16$ where ...
2
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2answers
50 views

Implicitly differentiate $e^y \cos(x) = 1 + \sin(xy)$

I can differentiate one side of the equation, but I dont know how to deal with sin(xy)
2
votes
3answers
40 views

Find the derivative of $\frac{(2x−1)e^{−2x}}{(1−x)^2}$

I need to find the derivative of $$\frac{(2x−1)e^{−2x}}{(1−x)^2}$$ I seems very complex to me so I'm wondering if there is a rule or formula I should be using? I attempted it using the chain rule ...
0
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3answers
37 views

Derivative Confusion

I am confused about something. In derivation we learnt that; a^x = a^x . lna Now the question that comes to mind is what is the difference when we have: a^3 =
0
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2answers
49 views

how to differentiate a function with square root

Trying to solve $y =7t^4-10 \sqrt {t+\frac{10}{t}}$ I know how to differentiate down to $7(4t^3)- . . .$ and I know a sqrt is equal to $x^.5$ but cannot figure out how to apply that to the rest of ...
-1
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1answer
64 views

Calculus - Finding the derivative [closed]

I am not sure how to solve this question: If $y = f(\sqrt{x^2+9})$ and $f'(5) = -2$, find the derivative of $y$ w/ respect to $x$ when $x = 4$.
0
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1answer
45 views

Nice proof for $\lim_{h\to 0}\frac{f(x+nh)-f(x)}{h}=nf'(x)$ besides LHR

Why is $$\lim_{h\to 0}\frac{f(x+nh)-f(x)}{h}=nf'(x)?$$ A cheap answer would be L'Hospital's rule, but I think there should be a more direct way to prove it, appealing to the first principles of the ...
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3answers
125 views

Max perimeter of triangle inscribed in a circle

What is the maximum perimeter of a triangle inscibed in a circle of radius $1$? I can't seem to find a proper equation to calculate the derivative.
2
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3answers
74 views

Equivalent definitions of differentiable

I am trying to show: The two statements are equivalent: (i) $f$ is differentiable at $a$, (ii) $f(a + h) = f(a) + ch + o(h)$, where c is some constant (depending on $a$) and $o(h)$ denotes some ...
3
votes
2answers
107 views

If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
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2answers
79 views

Evaluate $\frac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$ with logarithmic differentiation

Spivak asks us to evaluate $$\dfrac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$$ by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to ...
0
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1answer
39 views

Find Average and Instantaneous Velocity of a Function

Use the following function, f(t)=3t^3+t, to find the average velocity of: a. t=2 and t=0 b. t=2 and t=1 c. t=2 and t=1.9 d. t=2 and t=1.99 e. the instantaneous velocity at t=2 I have trouble with ...
0
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2answers
57 views

First derivative of $e^{-2x}/(1-x)$

Could someone please help me with these derivatives? I have to find the first derivative of $$f(x) = \frac{e^{-2x}}{1-x}.$$ Then I have to find the second derivative of that. For the first ...
1
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5answers
54 views

Finding the Derivative of a Derivative

Let $x^3+y^3=9$. Find $y''(x)$ at the point $(2,1)$. I keep getting $3x^2+(3y^2)y'=0$ as the first derivative then simplify that down to $-3x^2/(3y^2).$ But after that I keep getting ...
0
votes
2answers
92 views

Calculus Derivatives - Finding the slope of a function

I am having trouble solving this question: Consider the function $f(x)= 2x^{5/3} - 5x^{2/3}$ Determine the slope of the tangent at the point where the graph crosses the x-axis.
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3answers
75 views

Derivative of matrix and vector in $\mathbf {v^TMv}$

Suppose I have a ($n\times 1$) vector $\mathbf v$ and a ($n\times n$) matrix $\mathbf M$ and I want to compute the derivative w.r.t. some $x$. Both $\mathbf v$ and $\mathbf M$ depend on the scalar ...
-1
votes
2answers
248 views

Use implicit differentiation to find an equation of the tangent line to the curve

$$x^2+xy+y^2=3, (1,1)$$ I got the derivative as.. $$\frac{2x-2}{x+4}$$ But when I plug in the points I get the equation $y=x/2+2$ which is wrong. Is my derivative wrong? Or am I making a mistake ...
1
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1answer
33 views

Partial differentiation of a composite function

This should be straightforward, but don't seem to be able to crack it. Take a function $f(x_1, x_2, x_3)$ and a function $g(x4, x5, x6)$. These two functions mapp from $R^3 \rightarrow R^1$. I am ...
0
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2answers
121 views

Differentiable at $x=0$ only [closed]

Let $f(x) = x^2*1$ if $x$ is rational $f(x) = x^2*0$ if $x$ is irrational Show that $f$ is differentiable at 0 and not differentiable elsewhere.
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0answers
49 views

Help in differentiating a complicated function

How do I differentiate this function $u(x)=(\frac{1}{2}-\Pi (\epsilon ))u(x-\epsilon )+(\frac{1}{2}+\Pi (\epsilon ))u(x-\epsilon )$ twice with respect to $\epsilon$ and evaluate the derivatives at ...
0
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1answer
29 views

Why are these linear functions/operators? (Mathematical Methods… by Boas, Problem #3.7.13)

I had these questions on a homework of mine. My answers were marked incorrect, but I'm not sure why. Let $D$ stand for $\frac{d}{dx}$, $D^2$ for $\frac{d^2}{dx^2}$, $D^3$ for $\frac{d^3}{dx^3}$, ...
2
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2answers
71 views

Prove that $f'(x_0)=c$

Let $f:(a,b)\rightarrow \mathbb{R}$, and $x_0 \in (a,b)$. $f$ is differentiable at $(a,b)$. Also, Let $l(x)= cx+d$, "passes" at $(x_0, f(x_0))$. Prove that if $\forall x \in (a,b):f(x) \ge l(x)$ ...
0
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2answers
58 views

Cardinal curve - computing the bounding boxe

Problem description I have a cardinal curve, ie defined by the following basis functions : $h1(s) = 2 * s^3 - 3 * s^2 + 1$ $h2(s) = -2 * s^3 + 3 * s^2$ $h3(s) = s^3 - 2 * s^2 + s$ $h4(s) = ...
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0answers
3 views

Multidimentional Scaling with Pairwise distance “vectors”

Consider a random variable $z$ with a Gaussian distribution : $$ \mathbf{z} \sim \mathcal{N} ( \mathbf{m}, \mathbf{V} ) $$ Where $\mathbf{m}$ and $ \mathbf{V}$ are mean and variance parameters. ...
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0answers
27 views

Regularizing a function

I am working on an algorithm that requires derivatives of two functions that I need to take derivative of. Unfortunately they have sharp changes at two points so I have to regularize or smooth-en them ...
2
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2answers
115 views

Show that $f$ is not increasing on any interval containing $0$

$f:R\to R$, $f(x)=x^2\sin(1/x)+x$ if $x\ne 0$ and $0$ if $x=0$ In the first part of this problem, I showed that $f'(0)>0$ The second part of the problem is this: Show that $f$ is not increasing ...
1
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1answer
84 views

Why can't a non-zero polynomial satisfy some equations?

I'm having a hard time visually picturing/understanding how to explain why a non-zero polynomial function cannot satisfy the equation: $f''(x)$ = $-f(x)$ So is it basically asking to explain why a ...
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2answers
65 views

Vector derivative $\frac{d(Ax)}{d(x)}$ [closed]

I just need to know that whether it is $A$ or $A^T$ . I need it for an homework . Please be quick in telling me . Thanks !
0
votes
4answers
576 views

Derivative with a Square root in Denominator

$f(x) = \dfrac{-3}{\sqrt{3x^2 + 3}}$ I can't seem to figure this problem out. I think you would make the bottom(3x^2+3)^(1/2) and then use the chain rule on bottom and then use the quotient rule. ...
0
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4answers
67 views

Definition of second derivative as a limit

I found a statement that the second derivative can be defined as: $$\lim_{x \to a} \frac{f '(x)-f '(a)}{x-a}$$. Does this definion follow from the definition of the first derivative as: $$f ' (x) = ...
1
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2answers
248 views

Find the arc length of the curve $x = 1/6*(y^2+ 3)^{3/2}$ from $y = 0$ to $y = 1$

I am trying to find the arclength of the curve $$x = 1/6\cdot\left(y^2 + 3\right)^{3/2},\;\; 0\leq y\leq 1$$ I got this far and now I am stuck and don't know what to do next. Any help please? ...
0
votes
1answer
96 views

Verify by Second Derivative Test

$$A(x)=2\sqrt{x^2-16}+\frac14\sqrt{68x^2-x^4-256}\;,\;\; (4 < x < 8)$$ of which the derivative is: $$a'(x)=\frac{2x}{\sqrt{x^2-16}}+\frac{136x-4x^3}{8\sqrt{68x^2-x^4-256}}$$ I first had to ...
0
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2answers
150 views

Question about the differential

Today at class, my teacher stated the following proposition saying it is obvious: Let $x_0 \in U \subset \mathbb{R}^d$, $U$ open, and $f: U \to \mathbb{R}^m$ differentiable at $x_0$, then for any $v ...