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

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1answer
14 views

Continuous composed with differentiable

If $f(x)$ is $C^\infty$ and $g(x)$ is bounded and continuous does that imply that $f(g(x))$ is differentiable
3
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3answers
138 views

Implicit differentiation of $e^{x^2+y^2} = xy$

I just want to reconfirm the steps needed to answer this question. Thank you Find $\dfrac{dy}{dx}$ in the followng: $$e^{\large x^2 + y^2}= xy$$ I got this so far. ...
2
votes
1answer
40 views

derivative of sqrt(5/(x+7))

Why is it that: $$\frac{d}{dx}\sqrt{\frac{5}{x+7}} = -\frac{\sqrt{5}}2\frac{1}{(x+7)^{3/2}}$$ (image) ??? My attempt: It seems that somehow you end up adding 1 to 1/2 to get 3/2 in the exponent. ...
1
vote
1answer
31 views

Finding the tangent line through the origin

Find the tangent line to: $$f(x) = \sqrt{x-1}$$ that passes through the origin $(0, 0)$. $$f'(x) = \frac{1}{2\sqrt{x-1}}$$ The line will be tangent at $(a, b)$ so then: $$f'(a) = ...
-2
votes
2answers
56 views

Integral $\int x^7\cos x^4 dx$

$\displaystyle \int x^7\cos x^4 dx$ I tried first by letting $x^4 = u$ and then using integration by parts by assigning f(x) to $u^\frac74$ and cos(u) to g'(x) and I end up getting after applying ...
3
votes
4answers
59 views

$dx$ being a desginator (with respect to $x$) or being a term?

I am confused as to what $dx$ truly is. I am doing some u-substitution problems and this is what I came across: $$\int 2x(x-1)^{1/2}\,dx$$ $u=x-1$ and therefore $du=1$ when we substitute we get: ...
2
votes
2answers
51 views

derivative of $\sec^2(x/12)$

Alright, so the derivative of $\sec^2(x/12)$ is $\frac{1}{6} \tan\left(\frac{x}{12}\right) \sec^2\left(\frac{x}{12}\right)$ But if you use chain rule, you get: $$2 \sec\left(\frac{x}{12}\right) ...
2
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1answer
58 views

Example of non-differentiable continuous function with all partial derivatives well defined

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any ...
0
votes
2answers
32 views

differentiability and continuity in R3

Prove that if a function is differentiable at $(a,b,c)$ in $\mathbb R^3$ then it is continuous at $(a,b,c)$. I tried to imitate the proof that if $f$ is differentiable at a specific point in $\mathbb ...
1
vote
0answers
18 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-8)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
0
votes
2answers
43 views

Prove that $f$ has derivatives of all orders at $x=0$ [duplicate]

Let $\displaystyle f(x) = \begin{cases}e^{- \frac{1}{x^2}} &\text{for } x \neq 0 \\ 0 & \text{when } x=0 \end{cases}.$ Prove that $f$ has derivatives of all orders at $x=0$, and ...
1
vote
1answer
16 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-6)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
2
votes
1answer
19 views

Calculate Laplace transform of the product of t and f(t) by differenitating f(t) (5.5-4)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
0
votes
1answer
34 views

Differentiability in $\mathbb R^3$

$G$ is an open subset of $\mathbb R^3$ and $(a,b,c)$ belongs to $G$. $f$ is a function from $G$ to $R$. i) Define: $f$ is differentiable at $(a,b,c)$ ii) Prove if $f$ is differentiable at $(a,b,c)$ ...
0
votes
4answers
75 views

Derivative of $f(x) = x^5$ using the definition.

Let $f(x)=x^5,$ and $\quad P(1,1)$ $(a = 1,\text{ and } f(a) = 1)$. $$\lim_{h\to 0} \frac{f(a+h) - f(a)}h \implies\lim_{h\to 0} \frac{(1+h)^5 - 1}h=\lim_{h\to 0} \frac{1}h((1+h)^5-1)$$ After This ...
1
vote
3answers
127 views

Multivariable Calculus, rate of change.

An insect is moving on the ellipse $2x^2+y^2=3$ on the $xy$-plane in the clockwise direction at a constant speed of 3 centimeter per second. The temperature function $T(x,y)$ (experienced by the ...
1
vote
0answers
12 views

Linearlized curvature operator

While reading a paper, I came across the term for linearized curvature operator \begin{eqnarray} \kappa_1 = -\frac{1}{{(1+x^2)^{\frac{3}{2}}}}\frac{\,d }{\,d x^2} + ...
0
votes
1answer
54 views

How to find the minimum value of this integral?

I am struggling to find the solution to this problem. If anyone could help to explain how to solve this problem to me, it would be really appreciated. Let $$ f(x)=-\sqrt{3}x+(1+\sqrt{3}) $$ $$ ...
0
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3answers
82 views

Find 6th derivative of $(\cos(5x^2)-1)/x^2$ at $x=0$

Let $$ f(x)=\frac{\cos(5x^2)-1}{x^2} $$ We want to compute the $6th$ derivate of $f(x)$ at $x=0$. Using a calculator, I found $18750$ (which is correct). But I don't understand how to find this ...
0
votes
1answer
21 views

Lyapunov and Asymptotically stability

How do you determine if a function is Lyapunov or asymptotically stable? The definitions do not seem to tell us how to prove whether a solution is stable or unstable. For example, I am trying to ...
0
votes
1answer
56 views

How to formalize that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0 \implies$ $g$ “grows faster” than $f$?

I understand that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0$ implies that, for sufficiently large values of $x$, $f(x)<g(x)$, as a direct consequence of the definition of limit to ...
1
vote
1answer
46 views

Solve $h(x)+h'(x)(8-x)-32=0$ for x.

Solve $h(x)+h'(x)(8-x)-32=0$ for $x$.Where $$h(x)=\frac{\frac{1}{16}x^2 - 2 x + 80}{\left(\frac{1}{16}x^2 - 2 x + 20\right)^2}$$ Should I go with characteristic equations? or is there another way. ...
0
votes
1answer
36 views

What is the $n$th derivative of $\coth(x)$?

I would like to know the $n$th derivative of the Hyperbolic Cotangent, i. e., $\frac{\partial^n}{\partial x^n} \coth( x )$. So far, I have only found an expression for the $n$th derivative of the ...
0
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0answers
24 views

partial derivatives of a function with two variables.

I have $\dfrac{\partial x}{ \partial t}$ where $x=f(t,a)$. Can i express it as $\dfrac{\partial x} {\partial t} + \dfrac{\partial x}{ \partial a}$ ? Please help me . I am really stuck.
0
votes
1answer
69 views

Give the differential and the derivative of $f(X) = I − X(X^tX)^{-1}X^t$

I don't know what to do, maybe use the product rule. Give the differential and the derivative of the function $$f(X) = I − X(X^tX)^{-1}X^t $$
0
votes
1answer
48 views

Question on a special Derivative

I have this functional defined from a Hilbert space $H$, $J\colon H\rightarrow \mathbb{R}$ defined by: $$ J(u)=\frac12 \|u\|^2-\int_0^1(A(su),u) ds $$ where $A\colon H\rightarrow H$ is a potential ...
2
votes
3answers
56 views

General solution to ODE $ y''-Ay^5=0 $

What is the solution of $$ y''-Ay^5=0 $$ I got the solution $ y = {(3/4A)}^{1/4} x^{-1/2}$ using trial and error but how to solve this type of problem in general?
0
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1answer
31 views

An ant is walking up a hill. at what x does he see the blade of grass.

've been working on this problem with Mathematica and by hand-help with either would be fantastic. The blade of grass is given by the line segment from (32,1/5) and (32,8). The 2D hill is given by ...
1
vote
0answers
23 views

Determine lambda from a non-constant differentiable function of one variable

Suppose f is a non-constant differentiable function of one variable. Determine, with reasons, the value of $\lambda$ for which F(x, y) = f($\lambda x^{3}$ + y) satisfies the partial differential ...
0
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0answers
16 views

Calculating Taylor tasks (sinx)

Is there basically anything else behind this task except recognising x is Pi/4 and that it is awfully similar to sinx version of Taylor? First time posting so I probably made some administrative ...
1
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0answers
32 views

Iterations $F^n_h[f]$ of the operator $F_h[f]=D_h[f]\circ f^{-1}$

Let the $H$ be a collection of real valued invertible functions, define $f\circ g$ as composition, $f+g$ as the function $f+g(x):=f(x)+g(x)$ and define a family of functions $\{D_h\}_{h\in \Bbb ...
0
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2answers
23 views

Question about differentiability/continuity,please help

I was reading in my textbook that it says "a function $ f $ may have a derivative $ f' $ which exists at every point, but is discontinuous at some point." Before this there is a theorem that says ...
0
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0answers
41 views

continuity of tangential derivative across an interface

Suppose I know a scalar function $p(\mathbf{x})$, $\mathbf{x} \in \mathbb{R}^2$ or $\mathbb{R}^3$, is continuous across an interface, some curve $\Gamma$ in the domain. Denote the value of $p$ on ...
0
votes
4answers
97 views

How do you define the derivative of a function without an argument?

So the derivative of $f: x\mapsto f(x)$ is defined by $f':x \mapsto \lim_{h\to0}\dfrac{f(x+h)-f(x)}{\phantom{f}(x+h)\,-\,(x)}$. But is there a way to define $f'$ solely in terms of $f$, without ...
1
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0answers
22 views

Proof that second Frechet derivative is symmetric?

Is there a "nice" way to prove that the second Frechet derivative of a function between normed spaces is symmetric? Any proofs that I've managed to find seem quite messy and don't really give any ...
1
vote
1answer
67 views

Interesting Question about a derivative proof

I recently searched around SE, and found: How to solve this derivative of f proof The answer is interesting. "A function given that $f(x)=f''(x)+f'(x)g(x)$ could be an exponential function, ...
-1
votes
1answer
30 views

Partial derivative of a first order condition in microeconomics

Im currently studying microeconomics and I have encountered a math problem which I can't seem to figure out. The concerned problem can be viewed in the image that I have posted. To be more specific, ...
0
votes
1answer
53 views

About the function $f(x)=\sin x\ln x^2$ and derivative definition

$f(x)=\begin {cases}\sin x\ln x^2 & x\neq 0\\ 0 & x=0\end{cases}$ When I try to find the derivative on $x=0$ with the defintion I get: $\displaystyle\lim_{h\to 0}\frac ...
0
votes
2answers
52 views

Differentiable function- prove that there exists a point such that $ f'(\lambda)=0 $

Suppose that $ f:(I)\rightarrow R $ is differentiable and show if $ f(x)=f(y)=0 $ for $ a<x<y<b $ then there exists $ x < \lambda<y $ such that $ f'(\lambda)=0 $. I was thinking to ...
3
votes
3answers
72 views

Prove $f(x) = \frac{1}{x}$ is smooth (infinitely differentiable).

I have never proved that a function is smooth (infinitely differentiable) before. The only function that comes to mind which is smooth is $g(x) = e^{x}$, because it is defined on all of $\Bbb R$, ...
4
votes
2answers
107 views

Why does the derivative rule $a^x = \ln(a)\cdot a^x$ fail for $e^{-x}$?

In my textbook there is a derivative rule stated as folows: $$f(x)=a^x \implies f'(x)=\ln(a) \cdot \ a^x$$ But when I try to apply this rule to $e^{-x}$ I get: $$\ln(e) \cdot \ e^{-x} = e^{-x}$$ ...
3
votes
2answers
105 views

Show $f$ is not $1-1$

Let $f:\Bbb R^2\to \Bbb R$ be a continuously differentiable function. Show that $f$ is not $1-1$. I know I will need to use the Inverse Function Theorem and consider some open set A with $g:A\to ...
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1answer
47 views

Water leaking from box and the relationship of volume and height.

Suppose we have a container that has a base of area $b$ and we fill it up with water. Volume of water = $b \cdot h$, where $h$ is height. Hence, $\mathrm{d}v/\mathrm{d}t = b \cdot ...
0
votes
1answer
39 views

Divergence (or second derivative) of circle

The circle has the uniform shape because a second derivative is 1. That is an intuitive guess - the line turns around at constant rate (i.e. the first derivative changes at constant rate), which means ...
1
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0answers
61 views

Implicit equation

Problem: Let $f(x,y)=x-\ln x - y +\ln y$ for $x,y>0$. Prove that there exists a $\delta>0$ and a function $\varphi : (-\delta, \delta)\to \mathbb{R}:\ \varphi \in C^1,\ \varphi(0)<0,\ ...
6
votes
3answers
91 views

Proving that $\lim\limits_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$

Prove that $$\lim\limits_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$$ Is the following a correct proof: $f''(x)=$$\lim\limits_{h\to 0}\frac{f'(x)-f'(x-h)}{h}=\lim\limits_{h\to ...
1
vote
1answer
81 views

Derivative of trace of inverse matrix?

I've been trying to derive the formula for the derivative of $Tr(X^{-1})$ w.r.t. $X$, which I know is $X^{-2T}$. According to the Matrix Cookbook $$\dfrac{\partial g(U)}{\partial X_{ij}} = ...
1
vote
2answers
46 views

Derivative of the following function (similar to Softmax)

I am having a hell of time trying to differentiate the following function with respect to x. Do you have any suggestions $f(x) = \frac{ w(i)^x}{ \sum\limits_{j} w(j)^x }$ where $w$ is a vector ...
0
votes
2answers
30 views

Examples and graphs of functions that are once, twice, three times differentiable, etc.

I'm trying to deepen my understanding of differentiation and this idea of infinitely differentiable functions as being "smooth" -- i.e., the more a function is differentiable, the smoother it gets. I ...
1
vote
1answer
39 views

Finding a Lipschitz Continuous function in $D=[-1,1]$ that is not differentiable at all points in D

The problem is to find a Lipschitz Continuous function in $D=[-1,1]$ that is not differentiable at all points in D. To tackle this, I have considered functions I know to not be differentiable at ...