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
38 views

Analyzing if function is “onto”

I have some function $g$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $g: [a,b] \to [a,b]$. How can I find out if this is true or not? P.S. I am not saying all $g$ have the ...
-2
votes
0answers
26 views

detailed example of derivatives in our real life?? [on hold]

Derivatives are very important for lots of things especially in Physics and Engineering. I use derivatives almost every day as an engineer. In my work, I study vibrations of underwater pipelines. ...
0
votes
3answers
39 views

How do I find the critical points of this function involving e?

I have the function: $$g(x)={{1 \over \sqrt{2 \pi}} \cdot e^{{-(x-2)^2}/2}}$$ Through very tedious differntion, I got to: $$g'(x) = {{{-(x+2)} \cdot {e^{{-(x-2)^2}/2}}} \over {2 \pi}}$$ Setting ...
0
votes
2answers
31 views

Analyzing derivative of function.

I have some function $g: [a,b] \to [a,b]$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $\forall x \in [a,b]: |g'(x)| \lt 1$. How can I find out if this is true or not? P.S. I ...
2
votes
4answers
53 views

$n^\text{th}$ derivative of $\tan^{-1} x$

Find $$\frac{d^n(\tan^{-1}x)}{dx^n}$$ Or find the $n^\text{th}$ derivative of $\tan^{-1}x$ w.r.t. $x$. Differentiation 4-5 times did not patternize so as to find out the $n^\text{th}$ derivative. ...
4
votes
1answer
66 views

Why are the differentiation/integration rules what they are?

So I understand what rules you use where, and the general forms of the rules like: $$\left(\frac{d}{dx}\right)^nx^k=\frac{k!}{(k-n)!}x^{k-n}$$ My question is why are these the formulas that give us ...
2
votes
2answers
189 views

Differentiation Tricks

Since most derivatives are trivial to take, it's understandable why integrals get most of the mathematical tricksters' attention. However, not all derivatives are trivial to take and I think it's good ...
0
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2answers
30 views

At what argument $x$ is the tangent to the graph $y=\frac{1}{2}x^2-\ln x$ horizontal?

At what argument $x$ is the tangent to the graph $y=\frac{1}{2}x^2-\ln x$ horizontal? Well this is a question which I found in a website. I found the Derivative to be $(x^2+1)/x$. As far as I ...
-2
votes
2answers
62 views

Derivative $(1-x)^{-2}$ [duplicate]

I'm getting answer of $\frac{2}{(1-x)^3}$ , but online calculators suggest it's $\frac{-2}{(1-x)^3}$. I've tried it as $(1-x)^{-2}$, which results $-2*(1-x)*-1 = \frac{2}{(1-x)^3}$. Same result with ...
0
votes
1answer
33 views

The first assumption leads to the third one that looks inconsistent at a glance. Can you explain it better?

Background I am trying to solve the following problem: > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a$ such that ...
0
votes
1answer
56 views

Is an everywhere differentiable function locally Lipschitz?

If we have a differentiable function $ f:\mathbb{R}^n \to \mathbb{R}^n $, does it have to be locally Lipschitz? It's obviously true for continuously differentiable functions, but what happens without ...
8
votes
2answers
110 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
1
vote
1answer
68 views

Dimension of garden to minimize cost

Math question: A homeowner wants to build, along her driveway, a garden surrounded by a fence. If the garden is to be $5000$ square ft, and the fence along the driveway cost $6$ dollars per foot while ...
-5
votes
0answers
42 views

Find the values of $c$ that satisfy the Mean Value Theorem [on hold]

Find the value or values of $c$ that satisfy the equation $f'(c) = \frac{f(b)-f(a)}{b-a}$ in the conclusion of the Mean Value Theorem for the function and interval. $$f(x)= \ln(x-1), \ I = [2,6]$$ ...
0
votes
0answers
51 views

How to differentiate $y$ with logarithmic differentiation

I am asked to find the differentiate $y$ using logarithmic differentiation $$y=\frac{ x(x^5+1)^{1/2}}{(x-1)^{1/3}}?$$ I tried it 3 times and I got three different answer each time Any help
2
votes
1answer
33 views

Solve an initial value problem using the directional derivative

In my notes there is the following example of solving an initial value problem using the directional derivative. The problem is the following: $$u_t(x,t)=u_x(x,t), x \in \mathbb{R}, t>0 \\ ...
0
votes
1answer
35 views

Using log to take derivative of a function

Is it safe to say that if $\frac{d}{dx}ln(f)= g $ for some functions f and g, then $\frac{d}{dx}f = e^{g}$? Why or why not? (novice high schooler here)
0
votes
0answers
21 views

limit of recurrence serie by derive

how to define limit this series $$U_{n+1} = {df(U_n)\over dU_n}$$ $f(x)$ function derivable in Class $C^n$, $f^{(n)}(x)$ function not null and $U_0 \gt 0$
2
votes
3answers
48 views

Inequalities and Differentiation

Having become so accustomed to differentiation and integration being applied just like normal algebraic operators, I was somewhat suprised yesterday when I realized that $f(x) \geq g(x)$ does not ...
0
votes
1answer
19 views

Finding a solution for $(2\sin y-2x)y'-y=0$ that goes through the point ($\frac{2}{\pi},\pi$).

How do I find a solution for the differential equation: $(2\sin y-2x)y'-y=0$ that goes through the point ($\frac{2}{\pi},\pi$) by using the property of inverse function: $\frac{dx}{dy} = ...
1
vote
1answer
38 views

Understanding and teaching the concept of derivative

I need to prepare an introductory lecture about derivatives and the concept of differentiation to a class of people with a general mathematical background (who have also studied calculus a few years ...
0
votes
0answers
26 views

How to prove a function is negative over specified interval

I have a function $f$ as follows: $f(x)=\frac{((b-1)^2-x^4)}{x^2\sqrt{(x^2-b-1)^2-4b)}}+1$ where $b\gt0$ is a positive constant. I know that $f(x)\lt0 \text{ for } x\ge(\sqrt{b}+1)$ , but I don't ...
1
vote
0answers
18 views

If $f$ is derivable in $(a,b)$, proof that if $f=0$ in $k$ points, then $f'=0$ in at least $k-1$ points.

Let $f:[a,b]\longrightarrow \mathbb{R}$. Prove that if $f$ is derivable in $(a,b)$, proof that if $f=0$ in $k$ points, then $f'=0$ in at least $k-1$ points. I have to use Rolle's theorem for this, ...
-2
votes
0answers
14 views

Hessian of function of two norm [on hold]

I need to calculate the: ${\nabla ^2}f\left( x \right) = ? $ where $ f(x) = \gamma \left( {a,||x||_2^2} \right)$ and $ \gamma \left( {a,b} \right)$ is the upper bound incomplit gamma function. ...
3
votes
0answers
27 views

Derivative of a linear basis function over a moving mesh

Given a moving mesh $0=x_0(t)<x_1(t)<\cdots<x_N(t)<x_{N+1}(t)=1,$ where $t$ denotes the current time so that the mesh is moving with time. The linear basis function is then defined as ...
0
votes
2answers
25 views

In differentiability

Let $f(x,y)$ and $g(x,y)$ are differentiable functions in $x$ and $y$. Suppose $f(x,y) = F(g(x,y))$.My question, Is $F$ differentiable function?!.
-3
votes
1answer
35 views

Find the directional derivative using $f(x,y,z)=xy+z^2$. [on hold]

Find the directional derivative using $f(x,y,z)=xy+z^2$, at the point $(2,3,4)$ in the direction of a vector making an angle of $\frac{3\pi}{4}$ with grad $f(2,3,4)$. PS - I am having trouble ...
10
votes
2answers
117 views

Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)

In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...
1
vote
0answers
15 views

Gradient of the Fourier transform of a function

I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function. In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is ...
1
vote
1answer
25 views

Directional derivative vs. function restriction and then derivative

Say I have a function of two variables, and a line in the plane, and I'd like to "take the derivative along the line". Is this an indication to use the directional derivative, OR is it expected that I ...
-1
votes
3answers
57 views

What is the derivative of max and min functions? [on hold]

If I define a function: $f(x) = \max[g(x),h(x)]$ What is $f'(x)$?
2
votes
1answer
20 views

Differentiability of an absolute function.

Check the differentiability of $f(x)=x|x|$, $x$ is in $\mathbb{R}$. I know that it is differentiable when $x>0$ and $x<0$. I am not sure about the case when $x=0$. I found that as $$\lim ...
-2
votes
0answers
9 views

Use of Derivatives to find marginal revenue

The revenue function for a certain commodity is: $R(x) = 4 + \sqrt{3x+4}$ What is the marginal revenue when $x = 7$?
0
votes
2answers
21 views

Use ratio test to test for convergence or divergence

I have online hw and it tells me if my answer is correct or not. It said that my answer for this problem is incorrect: Can someone tell me what I did wrong? Also I might be asking alot of these ...
0
votes
0answers
19 views

Derivation: Discerning difference between arithmetic expression with parenthesis versus without using abstract syntax trees

I am trying to illustrate the expression: ( 3 * 4 + 5 * 6 + 7 ) using an abstract syntax tree. I have already illustrated the expression: ( 3 * (4 + 5) * (6 + 7) ). Could someone please illustrate ...
0
votes
1answer
23 views

How to demonstrate this?

I've a question and it is: Evaluate ${\partial^2z \over \partial u^2}+{\partial^2z \over \partial v^2}$, if ${\partial^2z \over \partial x^2}+{\partial^2z \over \partial y^2}=0$ and $z=z(x,y)$, ...
1
vote
2answers
40 views

Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.

I've been stuck on this question for a long time now and was wondering if anyone could show me how it's done. So far I have done the following: Since $\lim_{x\to \infty}f'(x)=M$ then $\forall \epsilon ...
1
vote
2answers
36 views

How do you find the general expression for the k^{th} derivative of an exponetial function with a function in the exponent?

I'm looking for a general expression for the function $\frac{\delta^k}{\delta \mu^k}[e^{n\mu + \mu^2}]_{\mu=0}$ I was thinking I could use the taylor expansion coefficients, but the function in the ...
1
vote
1answer
16 views

Cauchy–Riemann equation on differntiability

I have found that: $U_x = -\exp(y)\sin(x) $ $U_y = \exp(y)\cos(x) $ $V_x = \exp(y)\cos(x) $ $V_y = \exp(y)\sin(x) $ I need to show that $U_x=V_y$ and $U_y=-V_x$, however these aren't ...
2
votes
2answers
43 views

Proving the Derivative of cosine and sine functions

In the proof of the derivatives of cosine and sine functions, we used the facts that: $$\lim\limits_{\Delta x \to 0} \frac{\cos \Delta x - 1}{\Delta x} = 0$$ and $$\lim\limits_{\Delta x \to 0} ...
0
votes
2answers
29 views

harmonic function. How to prove?

I've with prove if $1 \over |x|$ is a harmonic function. I know with for a harmonic function, $f_{xx}+f_{yy}=0$, but I don't know how to derivate ${1 \over |x|} dx$. And I don't know how to derivate ...
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
2
votes
4answers
94 views

How to determine $\lim_{h \to 0}\frac{g(h+1)-g(1)}{h}$

It is given that $g(x) = x^{20}$ Determine $$\lim_{h \to 0} \frac{g(h+1)-g(1)}{h}$$ Can someone give me a hint please? I worked it out to be so far as: $$\lim_{h \to 0} \frac{(1+h)^{20}-1}{h}$$ The ...
2
votes
2answers
35 views

What derivative should be taken for relative maxima and absolute maxima (or minima)?

I get confused on what derivative should be taken for defining relative maxima and absolute maxima because some sources said to use first derivative while the others said to use second derivative. ...
-1
votes
0answers
32 views

Initial value problem with unique solution and rear wheel of a bike problem

Consider $I\subseteq\mathbb{R}$ an arbitrary interval (it can be of the following types: $[a,b], [a,b), (a,b], [a,+\infty),(a,+\infty),(-\infty,b), (-\infty, b], \mathbb{R}$, where $a,b\in\mathbb{R}, ...
0
votes
0answers
27 views

Osculating Circle in Differential Calculus

I am working with an osculating circle as the curve of closest contact to a curve in differential calculus and my book takes some confusing steps that I do not understand. It says: Let $f(x)$ be the ...
0
votes
2answers
41 views

Find the derivative of absolute value using the chain rule

I need help solving this derivative using the chain rule. I have tried setting $u = -x^2$
2
votes
5answers
79 views

How to differentiate $x^2-|x^3|$?

How to differentiate $x^2-|x^3|$? I tried breaking it into a piecewise function but I've been told this is not necessary. How can I approach this in another way?
0
votes
3answers
46 views

Derivative of a definite integral?

I could not figure out what I am doing wrong. Suppose $$f(x)= \int_1^x \sin(t^2) \ dt$$ What is $f'(x)$? I found $f'(x) = 0$. But it says this is not correct answer. Can someone please explain step ...
-3
votes
0answers
19 views

Critical point outside of domain when finding the intervals on which a function is increasing and decreasing.

I have this function: f(x)=x^(1÷3) × (x+8) I'm trying to find the intervals on which the function is increasing and decreasing. Then, I am to find the local extrema. I've done this: f'(x) = ...