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

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43 views

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

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
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0answers
56 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
36 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
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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
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1answer
21 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
48 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
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0answers
33 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
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0answers
19 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, ...
3
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0answers
28 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
27 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
37 views

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

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
121 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
17 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
26 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
61 views

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

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
11 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
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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
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0answers
22 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
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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
44 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
44 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
30 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
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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. ...
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0answers
34 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
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0answers
29 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
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2answers
42 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 $u = -x^2$
3
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5answers
83 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
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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
20 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) = ...
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votes
0answers
34 views

What would be my alpha and beta

I was given a question earlier that requires use of Taylors theorem $$\sum_{k=0}^{n-1}\frac{ (f^{(k)})(\alpha) } {k!}(t-\alpha)^k$$ The approximation function is $p(x) = 1 + x + \frac {x^2} 2$ The ...
1
vote
1answer
46 views

A $2$-variable non-differentiable function whose partial derivatives exist

If a $2$-variable function is not differentiable at some single point, but has finite partial derivatives for both variables at that point, can it also have a derivative in any direction, I mean is it ...
1
vote
2answers
71 views

Differential $dx$

I have some trouble understanding a thing. I will reproduce two texts from two different books. In the first, the author defines the function $T:\mathbb{R}\longrightarrow \mathbb{R}$, ...
0
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1answer
26 views

Derivative with summation operator

How do you take the derivative when there is a summation operator in this step.. $$\frac{d}{dt} \left[1-\sum_{n=0}^{k-1} \frac{(\lambda t)^n e^{-\lambda t}}{n!} \right] = \lambda e^{-\lambda t} ...
-1
votes
3answers
198 views

How to find the n-th derivative? [closed]

$f(x)=e^x\cos x$ Find the $f^{(n)}(x)$ without using comparison and find the $f'(x)$ without using mathematical induction. How to solve this question? Anyone can guide me?
12
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5answers
1k views

Are all continuous one one functions differentiable?

I was reading about one one functions and found out that they cannot have maxima or minima except at endpoints of domain. So their derivative , if it exists, must not change it sign , i.e. , the ...
1
vote
1answer
18 views

Differentiability of CDF at 0

This might seem to be a very trivial question but anyway here we go: I'm currently reading the paper "On the Value of a Random Minimum Spanning Tree Problem" by Frieze (1984) and I'm stuck on the ...
0
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1answer
39 views

Related Rates (part2) $y=\sqrt{x}$

$y = \sqrt{x}$ a) Find $\frac{dy}{dt}$ when $x=4$ , Given $\frac {dx}{dt}=3$ I did $$y = x^{1/2} \implies \frac{dy}{dt} = \frac{1}{2\sqrt{x}} \frac{dx}{dt}$$ then plugged in $x=4$ and $y=3$, but ...
0
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1answer
16 views

Rate of change of a multivariable equation w/ respect to another equation

So I was told to find the rate of change of $$ f(x,y) = x^2 − 3xy + y2 $$ with respect to $$r(t) = e^{2t}+t^2$$ I know usually I would take the derivative with respect to each variable and then ...
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1answer
15 views

How to prove $\oint_{\partial D} \left( u \frac{d u}{d x} dx - u \frac{d u}{d y} dy \right) = \oint_{\partial D}u\frac{d u}{d \mathbf{n}} ds$?

For some simply connected region $D \subset \mathbb{R}^2$ with boundary $\partial D$, length differential along boundary $ds$ and normal $\mathbf{n}$, and some sufficiently smooth function $u(x,y)$ ...
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0answers
23 views

Increase of the rms during diffusion - Derivatives calculation

We consider a diffusion equation of the form $$ \frac{\partial F}{\partial t} = \frac{\partial }{\partial x} \!\cdot\! \left[ - \mathcal{F} (t , x) \right] \, $$ where ${ \mathcal{F} (t ,x) }$ is the ...
1
vote
1answer
27 views

Finding $\frac{d}{dy} \int_{a}^{y} f(x,y)dx$ when $\int f(x,y)dx$ is non-trivial

I ran into a problem where I had to find the following $$\frac{d}{dy} \int_0^y \sqrt{x^4+(y-y^2)^2}dx $$ and was at a complete and utter loss as to where to begin. Any and all insights regarding ...
2
votes
1answer
38 views

How do I show that a directional derivative is defined in every direction (for every vector) for a function $f: \mathbb{R}^2 \to \mathbb{R}$?

I'm working through some analysis books, and while working through the section on directional derivatives, I searched here and found this answer, which states Let $f: R^2 \to R$ be defined by ...
1
vote
3answers
52 views

How to calculate $\frac {d}{dx}$ ($\int_{1}^{x^2} \sqrt{ln(t)}\,dt$), when $|x|>1$

How do I calculate $\frac {d}{dx}$ ($\int_{1}^{x^2} \sqrt{ln(t)}\,dt$), when $|x|>1$ ? After thinking about this I concluded that the operation $\frac {d}{dx}$ gets us back the original function. ...