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

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2answers
51 views

Simplification & Differentiation of $\frac{2x}{x^{1/3}}$

Above is the image I had taken a snap shot of. I was working on the problem # 24. I got to rewrite the function as: $y = 2x(x^{-1/3})$ I differentiated it and got the $y'$ as: $y' = ...
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2answers
35 views

Implicit differentiation with logarithm

Find $y'$ if $y=\ln(7x^2+3y^2)$. I'm kind of confused on this problem and could use everyone's help. Am I supposed to take the derivative first, which is $y=\ln(14x+6y)$? If so, how do I go from ...
0
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1answer
17 views

Scalar derivative of quadratic form where matrix depends on variable

I have the expression $$K(p(t),q(t)) = p^T D(q) p$$ Where D(q) is an n x n symmetric matrix, q and p are vectors (n x 1) depending on scalar variable t. I need to take the derivative of K with ...
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1answer
35 views

Finding Derivatives of Functions

I've recently been reading a text on classical mechanics and the Mathematics applied to the use of Derivatives and second derivatives, and I was hoping some folks could verify my answers to the ...
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1answer
42 views

How does this series expand the expression?

How does $$\sqrt{R^2 + |x|^2} = R + \frac{|x|^2}{2R}+\cdots$$ when expanded around the point $x=0$? I tried using a Taylor expansion but it didnt work out.
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2answers
36 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
6
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1answer
108 views

What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} ...
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2answers
45 views

Derivative at 4, when $f(x)=\frac{1}{\sqrt{2x+1}}$

Derivative at 4, when $f(x)=\frac{1}{\sqrt{2x+1}}$ I choose to use the formula $\displaystyle f'(x)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$ Which after some work I found to be ...
1
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2answers
43 views

Partial derivative in two dimensions

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 ...
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2answers
87 views

What's $\frac{dy}{dx}$ of function $y=\frac 12{\sin x}$?

$(1)\quad $ How to differentiate $y=\frac 12 \sin x$? I know that $\frac{dy}{dx}$ of $y=\sin x$ is easy to calculate: $\frac{dy}{dx} = \cos x$. But what if there is a coefficient preceding before it? ...
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0answers
17 views

Using Leibnitz Theory to find 2nd order differentiation of an equation.

Here is my solution but I am getting stuck at a later stage. Can someone point out my mistake. $ f(x)= x^2log x $ Let $u=log x$ and $v=x^2$ We know $u_n = (-)^{n-1} .{(n-1)!}. (x)^{-n}$ Using ...
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0answers
10 views

second derivatives and coordinate transformation [closed]

This is my problem. If (x, y) =f1 (u, v) and N=f2 (x, y) Then {∂N/∂x; ∂N/∂y} =J^-1 {∂N/∂u); ∂N/∂v} Here J= [∂x/∂u ∂y/∂u ∂x/∂v ∂N/∂v]; Similarly I need to calculate the relation between ...
1
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0answers
38 views

Does differentiability imply continuity for a derivative? [closed]

If $f(x)$ is differentiable at a point $c$, then is $f'(x)$ continuous at $c$? If so (or if not,) please provide a proof or a counterexample.
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0answers
17 views

Deriving maximum value from the Maxwell-Boltzman equation

I'm having some trouble with my math here. The Maxwell-Boltzman equation is: $F(v)=4\pi v^2(\frac{M}{2\pi RT})^{\frac{3}{2}}e^{-{\frac{Mv^2}{2RT}}}$ I begin by $f(v)'=4\pi (\frac{M}{2\pi ...
0
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0answers
15 views

Checking this second derivative

Is this correct? $$n'(\theta) = \frac{f'(\theta) - \alpha}{c'(n(\theta))}$$ $$n''(\theta) = \frac{c'(n(\theta))[f''(\theta)-[f'(\theta)-\alpha][c''(n(\theta)][n'(\theta)])}{\left [ c'(n(\theta)) ...
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2answers
30 views

Replace x with -1 in $ y' = \frac 32({1+x^\frac 23})^\frac 12 \ ({\frac 23 x^\frac {-1}3}) $

Question: Find an equation of the tangent line to the given curve at the given point $ y = ({1+x^\frac 23})^\frac 32 \ $ at $ x = -1$ This gives the slope: $ y' = \frac ...
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1answer
28 views

Calculus - Implicit Differentiation

I'm reading math notes online here. In the notes there is a problem that differentiates the following equation: $$ \sec(A) = \frac x {50};$$ ...where the angle $A$ is a function of time (ie. $A = ...
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1answer
15 views

How to find a function with two variables from two functions with one variable

I am trying to determine a function for an algorithm I wrote. The time $t$ it takes to run depends on two variables $w$ and $l$ (with $l > 0$ and $w > 0$) I measured $t$ with a fixed $w$ ...
0
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1answer
93 views

Tangent Line Question

Please help with this question -urgent! thanks! In this problem we consider drawing some straight lines which form a nice pattern. Consider joining the point (0.1,0) to the point (0,0.9) by a line ...
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2answers
29 views

Basic question about limits and derivatives

the limit $\lim_{h \to 0}\dfrac{\sqrt{81+h}-9}{h}$ represents the derivative of some function $f(x)$ at some number a. Find $f$ and $a$. I don't quite understand what this question is asking. Is ...
1
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1answer
16 views

use derivative to get max surface of block

We have a block (a*b*c) with volume of 1 m^3. There are 2 questions: write surface area of block as function of a and b find a and b in a way so that surface will be as big as possible I had ...
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0answers
13 views

Finding all directional derivatives of a function involving absolute value.

I need to find all the directions in which the directional derivative exists for the function f(x,y)=|2x+y| at the point (0,0) and their values. So I used: $$ D_v(f)=\lim_{t\to 0}\frac {f(0+tu) - ...
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2answers
161 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
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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
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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 ...
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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
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2answers
73 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 ...
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1answer
25 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 ...
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2answers
35 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 ...
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2answers
26 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 $f(x)=-x^2+3x$, ...
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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 ...
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4answers
86 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
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0answers
56 views
+50

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 ...
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0answers
26 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 ...
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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 $\;\dfrac{dy}{dx}$.
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0answers
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+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 ...
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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 ...
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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
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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 ...
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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.
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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 ...
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2answers
28 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.
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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 ...
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0answers
12 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 ...
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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 ...
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0answers
22 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 ...
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2answers
39 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
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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
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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} ...
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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 ...