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

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-9
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2answers
210 views

Derivatives and calculus [closed]

How do I know if $f'(x)$ exists, is there any kind of rules for this?, If anything could you please tell me the rules
1
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2answers
37 views

Derivative of a function when it is squared.

Was wondering when you are for example finding $dw/dt$ but you are given a function like $w^2(t)=r^2-2\cos(t)$, when r is some constant, how you are supposed to solve it? Are you supposed to ...
0
votes
1answer
28 views

Partial derivatives exist, but the function is not differentiable

It is well-known that a function $f:\mathbb{R}^n\to \mathbb{R}$ can have the property that it is differentiable along any line through the origin, but not even continuous at the origin. Can the same ...
0
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1answer
28 views

Question about limits of a convex function

Is the following conjecture true? Suppose $f'(x)>0$ and $f''(x)>0$ for all $x \in \mathbb{R}$. Moreover, $\lim_{x \to -\infty} \frac{f''(x)}{f'(x)} > 0$. Then $\lim_{x \to -\infty} f(x) > ...
2
votes
2answers
74 views

Is there any geometric explanation of relationship between Integral and derivative?

It is said integral is anti-derivative, derivative is tangent of graph function in each point on the function and integral is the area of the region in the xy-plane bounded by the graph. I can not ...
1
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1answer
26 views

slope of a line question.

Let $f(x)=(1/3)x+10$ where $f$ models the number of people joining a badminton club after $x$ years from starting. Now the slope of $f$ is $1/3$, so that means that people are joining the club at a ...
3
votes
2answers
37 views

Properties and notation of third-order (and higher) partial-derivatives

This question has been bothering me for quite a while and I still haven't found a satisfying answer anywhere on the internet or in any of my books (which may not be that advanced, mind you...). Since ...
1
vote
2answers
27 views

Factoring the Negative outside of Parentheses Squared

Dealing with the four step process of finding a derivative. I haven't been in math for a year and a half, so I've forgotten a lot of basic rules. So basically, if the function is $f(x)=-x^2+3x$, ...
0
votes
1answer
35 views

Finding the slope at two points.

I have been sitting at this for 2 days and I'm not getting anywhere, admittedly I might be just very dumb when it comes to mathematics, and as such I would really appreciate some help with this. I ...
0
votes
4answers
87 views

How do I differentiate polynomials

can someone show me how to differentiate stuff like x + 2 and I've never did this before and I use the most god awful textbook imaginable. Much thanks
7
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2answers
79 views

Modified Hermite interpolation

I asked similar questions here and here, but I tried to formulate this one in a sharper way. Is anyone aware of some literature on polynomial interpolation where we sample the function and its ...
0
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0answers
29 views

A converse proof that involves Torsion, curvature, and differentiation that equates to 0

I am having difficulty proving the converse in part B. I understand part A and have shown that t/k-t/k = 0. I found that n=-1/k, n'-bt= b'+tn = 0, so n' = bt and b'=-tn. However, I am unable to find ...
-1
votes
2answers
51 views

Finding $\frac{dy}{dx}$ given $y= \frac{ \sin x + x^2 }{ \cot 2x}$

I am unable to differentiate the following: Given $\;y= \dfrac{ \sin x + x^2 }{ \cot 2x},\;$ find $\;\dfrac{dy}{dx}$.
3
votes
1answer
131 views
+50

Problem with notation in a thesis

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
4
votes
6answers
80 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
2
votes
3answers
89 views

How to find the derivative of the inverse function $g^{-1}$, when no formula for the function $g$ is given?

If $g$ is a strictly increasing function such that $g(7)=3$ and $g'(7)=7$, find $(g^{-1})'(3)$. I'm not saying to just give me the answer. I want to understand what the problem is asking and how ...
1
vote
1answer
18 views

An upper bound for $f$, given a differential inequality

Let $f : \mathbb R \to (0, \infty)$ be a function (At least a locally Lipschitz function) so that $$f + f' \leq C$$ for some positive constant $C$. Does that imply $f\leq C'$ for some $C'$? Of ...
0
votes
1answer
23 views

Differentiating this trigonometric function

Differentiating $$ L = \frac{2v_o^2\cos^2\theta}{g\cos\alpha}\cdot(\tan\theta-\tan\alpha) $$ with regard to theta. I know I have to use trig. idendities, but I'm just completely stuck.
1
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1answer
30 views

Estimate value using Lagrange's MVT

Estimate the value of $51^{1/2}$ using Lagrange's MVT. Answer both in terms of inequalities and approximately estimated value. My method: Let $f(x)=x^{1/2}$ defined in $[49,51]$ and ...
1
vote
2answers
30 views

Can anyone explain how to differentiate the Lambert W function?

I'm interested in the differentiation of the Lambert W function $y = xe^x$. I am unable to understand how to proceed for it.
1
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1answer
44 views

First and second derivatives of the function $f(x)=x\int_0^x e^{t^2}dt$

I haven't done calculus for a while so I need your help with these two exercises. I am not sure whether my solutions are correct so I'd really appreciate someone's feedback. $$ f(x)=x\int_0^x ...
0
votes
0answers
13 views

Exponential and Logarithmic Differentiation.

Q. If $xe^{xy}=y+sin^2x$, then find $\frac{dy}{dx}$ at x=0. If we differentiate the function directly as follows: $e^{xy}+xe^{xy}\left[y+x\frac{dy}{dx}\right]=\frac{dy}{dx}+sin\left(2x\right)$ At ...
0
votes
0answers
12 views

Derivative of double-dot product of tensors

I need to obtain the following expression: $\frac{\delta F}{\delta a}$ Where $a$ is a second-order tensor and $F=\frac 23 a_{ij}a_{ij}$ So that a double-dot product (or double contraction) of ...
1
vote
1answer
28 views

Estimate the difference between $f$ and $p$ interpolating $f$

Suppose $p$ is the unique polynomial of degree $\leq 2$ that agrees with a function $f$ at points $a_1 < a_2 < a_3$. If the third derivative $f^{(3)}$ exists, and $x\in (a_1,a_3)$, then we can ...
2
votes
2answers
40 views

What is the derivative of $\dot{x} = f(x(t))$?

I am supposed to take the derivative of a function similar to this one: Take the derivative of $$\dot{x} = \cos(x)$$ where $x$ is a function of $t.$ I believe that this can be generalized to the ...
0
votes
0answers
22 views

Prove n'(c) =0 with Mean Value Theorem

a. I could use f(b) =78 and f(a)=78 where b=10 and a=0 to prove that there is at least one value 0 but the problem says that x=c be some value such that c cannot =0 nor 10. So I can't use a=0 and ...
1
vote
1answer
40 views

Differentiation Proving

Can someone please help me solve this question. Provide a hint? If $$\cos\frac x 2\cos\frac x 4\cos\frac x 8\cdots=\frac{\sin x}x$$ then prove that $$\frac{\sec^2(x/2)}4 + \frac{\sec^2(x/4)}{16} ...
1
vote
3answers
59 views

Find the derivative of a function with an integral [closed]

So I have this question to solve. Given that:$$t = \int\tan(x/2)dx$$Find: (In terms of t)$$ \frac{dt}{dx}$$. Edit: I believe that the book is missing the integral sign, however, I am still confused ...
1
vote
2answers
37 views

$n$th derivative of $e^{-x^2}$

I observed that $f^{(n)}(x)= \begin{cases} e^{-x^2} & \text{if $n=0$}\\ -2xe^{-x^2} & \text{if $n=1$}\\ f^{(n-1)}(x)-f^{(n-2)}(x) & \text{otherwise.} \end{cases}$ How to get the closed ...
0
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1answer
18 views

Differentiating by Partial Differentiation.

Among the methods for finding derivatives, differentiating by partial differentiation looks interesting. Is there any general proof for this method. For instance my text mentions this method. Let ...
0
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0answers
39 views

Partial differentiation

I am able to get the hint. But I still could not get it how could I find the two partial differential. Maybe you could show my the first one and I maybe could get it. For this question, I could not ...
1
vote
2answers
67 views

$\left|f'(x)\right| \leqslant g'(x)$ implies $\left|f(x)-f(a)\right| \leqslant \left|g(x)-g(a)\right|$

Suppose that both $f$ and $g$ are real, differentiable functions over $[a, +\infty)$, where $a$ is a real number; and that for all $x \geqslant a$, $\left|f'(x)\right| \leqslant g'(x)$. Prove that ...
1
vote
1answer
84 views

A polynomial agreeing with a function and its derivatives

If we want $$p(x_i)=a_i, \qquad x_1 < \dotsb < x_{n+1},$$ then there is a unique polynomial of degree $\leq n$ that accomplishes this (Lagrange interpolation). If we want $$p(x_i)=a_i, \qquad ...
0
votes
2answers
34 views

How does the graph shows horizontal tangency,but how?

. I am supposed to find critical numbers from this graph. I have read the solution. It says to find the critical numbers I have to find to horizontal tangency of the graph. I know that horizontal ...
0
votes
0answers
23 views

How to differentiate an expression involving big-o notation?

From Apostol - Introduction to analytic number theory (Theorem 3.3) we have $$ x\geq1, \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}):=E(x), $$ I want to differentiate $E$ -- to get a rough ...
0
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1answer
39 views

Is my calculation right for differentiability?(with complete resolution if right)

In the following completed example I ask if it is done right. $$f(x,y)=\begin{cases} \frac {2x^2y}{x^2+y^2} \mbox{for} (x,y)\neq (0,0) \\0 \mbox{for} (x,y)=(0,0) \end{cases} $$ Now and the partial ...
1
vote
2answers
18 views

Not sure how to differentiate implicitly using parametric equations…

I am not sure (not taught before explicitly) how to apply implicit differentiation on parametric equations when I am solving the question posted below. Question Two positive numbers $x$ and $y$ ...
1
vote
2answers
59 views

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then ...
1
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0answers
30 views

Does linearity imply differentiability?

Is a linear function differentiable at every point by definition?
0
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3answers
46 views

limitations problem needing help to solve

$$\lim_{k\to x}\frac{ \sin^2 k - \sin^2 x}{k - x}$$ I have tried to solve it over and over but couldnt. I will be very happy if someone can show me the way to solve this .
1
vote
1answer
95 views

What is the (rigorous) reason that the derivative of $|x|$ does not exist at $x=0$?

Let $g=|x|$. Then, the derivative at $c=0$ is given by: $$ g'(0) = \lim_{x \to 0} \frac{|x|}{x} $$ which is either $+1$ if $x$ comes from the positive $x$-axis or $-1$ if $x$ comes from the negative ...
0
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6answers
71 views

How do you find the derivative of $2^{\sin(\pi x)}$?

I don't understand how to take the derivative of this expression. $$y=2^{\sin (\pi x)}$$
0
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2answers
144 views

Differentiation with respect to variable

After scouting around I've had no luck in finding answers to my question which is relatively simple for alot of you. Before jumping straight to the question I 'd like to clarify that I'm not only ...
-1
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1answer
21 views

Application of a derivative [closed]

A spherical projectile 40 cm in diameter and weighing 32kg is shot directly upward from ground level at 196m/sec. Ignoring air resistance during its flight, what is the max height the ball will ...
1
vote
3answers
145 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
1
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0answers
39 views

Derivative : tangent line and multiplication of derivative

Could anyone give me a hint how to prove the following statements ? I suppose that I have some general ideas of pictures of functions satisfying the condition should look like. But it seems impossible ...
0
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0answers
20 views

Error of Riemann sum is $a/n + o(1/n)$ [duplicate]

A problem from an old qual: For $f$ of class $C^2$, find $a$ such that $$\int_0^1 f(t)dt-\frac1n\sum_{k=1}^{n-1}f\left( \frac {k}{n}\right)=\frac{a}{n}+o\left(\frac1n \right).$$ If we divide ...
0
votes
1answer
31 views

how the ratio of two functions change…what am I doing wrong?

For $s \in \{1,\dots,T-2\}$, let $g(s) := \frac{f(s+1)}{f(s+2)} = \frac{\sum_{t=s+1}^{T} \frac{0.99^{t-1}}{1 + \text{exp}\left(\frac{t-1}{3} - 9\right)} \frac{1}{t}}{\sum_{t=s+2}^{T} ...
1
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2answers
39 views

Calculus on Matrices [closed]

I have a basic doubt regarding calculus involving matrices. Dimensions of each matrices are also indicated along matrix name Question If I have a matrix $\kappa(s)_{3\times 1}$ what is ...
1
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3answers
37 views

Differentiating a function by simplification.

If we consider a function: $f\left(x\right)=\dfrac{x-1}{2x^2-7x+5}$ This function is not defined at x=1 and x=5/2. So if we differentiate this function by u/v method we have: ...