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
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
41 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
votes
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
votes
2answers
50 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
votes
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
votes
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 ...
1
vote
3answers
129 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
votes
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. ...
0
votes
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
votes
1answer
40 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
votes
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
vote
5answers
56 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.
0
votes
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
257 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
vote
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
votes
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.
0
votes
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
votes
1answer
30 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
votes
2answers
72 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
votes
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) = ...
0
votes
0answers
4 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. ...
1
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0answers
28 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
votes
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
vote
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 ...
-2
votes
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
596 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
votes
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
vote
2answers
253 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
97 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
votes
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 ...
0
votes
0answers
27 views

Proving local minimum property

f is twice continuously differentiable on [a,b], f '(c)=0 for some c, and f "(c)>0. Prove that f(x)>f(c) for $x \in (c-\delta, c+\delta)$ such that $x \neq c$, so that x is a local minimum. This ...
1
vote
1answer
58 views

Neighborhood of a differentiable function [duplicate]

Show that if f is differentiable on a neighborhood of $[a,b]$ and $f'(a) < m < f'(b)$ then there exists $a$ in $(a,b)$ such that $f'(c) = m.$ First off, what is a neighborhood of an interval? ...
0
votes
3answers
31 views

Differentiation wrt to L

I need to differentiate a equation which I have some problem with. The equation looks like this: $(K-L)(x/L)^{\gamma}$. I need to differentiate this wrt to L. Not able to do it. Need some ...
1
vote
0answers
25 views

Conclusions from derivative

Suppose f is continuously differentiable on [a,b] and f(a)=2 and $|f'(x) \leq 0.3|$ for all x. What can you say about f(b)? The only result I could get was $\frac{f(b)-2}{b-a} \leq 0.3$ by the mean ...
0
votes
1answer
49 views

Proof with mean value theorem

I am having trouble completing this proof. Prove that $$\lim_{x \to 0} \frac{\cos x-1}{x}=0$$ using the mean value theorem. The mean value theorem guarantees that we have a c such that ...
0
votes
1answer
47 views

Differentiability of trigonometric piecewise functions

So I have a function of a real variable $x$: $f(x) = \left\{\begin{array}{lr} x \int_0^{tanx} \dfrac{t^2}{\sqrt{1+t^3}}dt & if \: x \ge 0\\ sin^2(x) & if \: x \lt 0 ...
2
votes
3answers
72 views

Function which is never its own ($n^{th}$) derivative?

Is there any real-valued function $f(x)$ of a real variable $x$ with $n^{th}$ and $m^{th}$ derivatives never equal for nonequal nonnegative $m$ and $n$ and where the $n^{th}$ derivative of $f$ never ...
0
votes
4answers
29 views

Proving differentiability with inequality

Given: $0 \leq f(x) \leq x^2$ for all $x$. Prove that $f$ is differentiable at $ x=0$, and find $f '(0)$. Give a counterexample of a function which satisfies the hypothesis, but which is not ...
1
vote
3answers
71 views

First and second derivative

The function $$f:\mathbb{R}\rightarrow (0,\infty)$$ is twice differentiable with $$f(0)=f'(0)=1$$ and $$f(x)*f''(x)+(f'(x))^2=1-\sin x$$, for all real numbers $x$. I have to prove, that ...
2
votes
1answer
70 views

Does the derivative of $\;\sqrt{x}- \arctan \sqrt{x} \;$ exist at $x=0$?

This is an exercise in Apostol Calculus Vol.1 E6.22 Q.16. Let $\,f(x)=\sqrt{x} - \arctan \sqrt{x}.\;$ Then $$\begin{align} f'(x) &= \frac{1}{2 \sqrt{x}}- \frac{1}{1+(\sqrt{x})^2} \frac{1} ...
1
vote
1answer
20 views

Take the derivative of this likelihood function

$\displaystyle L=-\frac {n}{2}\log(2\pi \sigma^2)-\frac {1}{2\sigma^2}\sum_{i=1}^n(Y_i-\mu_y)^2$ Take the derivative with respect to $\sigma^2$ and $\mu_y$. $\displaystyle \frac {\partial ...
0
votes
4answers
42 views

Derivative of 3 functions using the product rule

$y = (x + 1)^{10}(2x + 3)^{11}(4-x)^{12}$ Using the product rule and the power rule I get to this $ \frac{dy}{dx} = 10(x + 1)^9 (2x + 3)^{11} (4-x)^{12} + \\ 11(x + 1)^{10} (2x + 3)^{10} ...
0
votes
1answer
49 views

Second Order Differential Equation Question

Got this question on my FP3 homework - if anyone could help me out I'd really appreciate it. .
4
votes
2answers
112 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
0
votes
2answers
50 views

About the number of solutions of $a\log(x)=x^2$

I was wondering if you could help me with that problem: Find the solutions of\begin{equation} a·\log(x)=x^2, \end{equation} according to the values of $a\gt 0$. I thought that I could write ...
0
votes
0answers
20 views

The radius of a circle grows at a rate of $ 30$ cm/s, that rate increases the area of ​​the circle with respect to time?

I would like to address the question The radius of a circle grows at a rate of $ 30$ cm/s, that rate increases the area of ​​the circle with respect to time? I know I have to derive, but where I ...
0
votes
1answer
63 views

How do you differentiate a function with respect to the negative of its variable?

How do you differentiate a function with respect to the negative of its variable. For example, is it true that df(-x)/dx = - df(x)/dx? If so, why is it?
1
vote
1answer
143 views

Counterexample for mean value theorem

Using Lagrange's mean value theorem. We have $$\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=f'(x_0+\theta \Delta x)$$ Taking limits on both sides, we see that the limit of the derivative always equals to ...