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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2answers
40 views

differentiability at a point (0,0) based on partial derivatives

For $$ f(x,y)=\begin{cases} y^2 sin\left(\frac{x}{y}\right) & \text{if } y\neq0 \\ 0 & \text{if } y=0 \end{cases}$$ i've shown that it is continuous and that the partial derivatives ...
0
votes
0answers
139 views

Differentiation of an integral with lim

I'm looking at a proof that wants to show for $d\geq3$, $y\in\mathbb{R}^d$ and $x\in B_r(0)$ $$\lim_{h\rightarrow 0} \int_{B_r(x)} \frac{y_j}{h}\left(\frac{1}{|y-hu_i|^d}-\frac{1}{|y|^d}\right) ...
3
votes
2answers
369 views

Find $f'(8.23)$ where $f(x)=23|x|−37\lfloor x\rfloor+58\{x\}+88\arccos(\sin x)−40\max(x,0)$

Let $$f(x)=23|x|−37\lfloor x\rfloor+58\{x\}+88\arccos(\sin x)−40\max(x,0).$$ Find $f^\prime(8.23)$. Note: For a real number $x$, $\{x\}=x−\lfloor x\rfloor$ denotes the fractional part of x. I don't ...
0
votes
1answer
30 views

How to evaluate the derivative of a function defined by different formulas in different intervals?

Let $f$ be a function defined as following: $f(x) = e^{\frac{-1}{x^2}}$ if $x \neq 0 ; f(0) = 0$ How should I proceed to evaluate it's derivative in $x= 0?$
2
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1answer
97 views

Can the chain rule be proven by math induction?

I need to prove the chain rule for a math project and I am wondering if it can be proven by math induction. If not, how can this rule be proven?
0
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2answers
48 views

Finding Derivative $f(x)=|x-3|$

I need to find the derivative from the right and the left like, $$f'_-(x) = \lim_{\Delta x \to 0^-} {f(x_1 + \Delta x) - f(x_1) \over \Delta x} \\ f'_+(x) = \lim_{\Delta x \to 0^+} {f(x_1 + \Delta x) ...
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1answer
23 views

Operator on polynomials, antiderivative

We are given a linear map: $$\mathbb{R} [X] \ni p \rightarrow q \in \mathbb{R} [X], \ \ q'=p, \ \ q(0)=0 $$ and two norms on $ \mathbb{R} [X]$ : $||p||_{\infty} = \sup _{t \in [0,1]} |p(t)|$, ...
0
votes
3answers
77 views

Derivative of the function $f(x) = |x+2|$

How can I get the equation of the derivative $f(x) = |x+2|$ ? I have already graphed the original function $|x+2|$ and the derivative function, but I'm not sure how to find the derivative, the ...
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3answers
43 views

Need help with inverse trig functions

How can I prove $\arccos\frac{1-x^2}{1+x^2}$ is equal to $2\arctan x$ for $x\geq0$ ? I am also supposed to use the fact that if a function is defined and differentiable in $(a,b)$ and $f '(x) = 0 ...
1
vote
0answers
25 views

Term for a Convex Function whose derivative is also convex

Let $f(x)$ be a monotone non-decreasing convex function such that its derivative $\frac{d}{dx}f(x) = f'(x)$ is also a convex function. Is there a term in literature that is used to refer to such ...
-1
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1answer
18 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
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votes
3answers
139 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
44 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
56 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
57 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
52 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
votes
1answer
121 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 ...
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2answers
35 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
19 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
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2answers
48 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
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1answer
17 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
78 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
156 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
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0answers
13 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
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1answer
100 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
94 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
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1answer
31 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
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1answer
67 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
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1answer
52 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
40 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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1answer
71 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
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1answer
50 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
58 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 ...
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0answers
25 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
17 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 ...
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0answers
34 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
34 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
49 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
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4answers
98 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 ...
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0answers
30 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
84 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, ...
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votes
1answer
35 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
56 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
62 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
86 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
111 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: $$\ln(e) \cdot \ e^{-x} = e^{-x}$$ ...