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

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3answers
41 views

Differentiating $\int_{\cos x}^{4x}\cos u^5 du$

Use part one of the Fundamental Theorem of Calculus to find the derivative of $$\int_{\cos x}^{4x}\cos u^5 du$$ My answer: $$y\;'=\sin x (\cos^5(\cos x))+4\cos^5(4x)$$ What am I doing wrong?
1
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2answers
40 views

Finding derivative of $F(x)=\int_{x^3}^{x^6}(2t-1)^3dt$

Find the derivative of the following function using the Fundamental Theorem of Calculus: $$F(x)=\int_{x^3}^{x^6}(2t-1)^3dt$$ I don't know how to do this problem using FTOC because this is not ...
2
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0answers
57 views

solving this differential equation for $y$, Is it even possible?

Lets say I have the following: \begin{gather} \frac{(y')^3 + 3 y' y'' + y'''}{(y')^2 + y''} = \sqrt{1+(y')^2}\\ \frac{((y')^3+3y' y'' + y''')^2}{((y')^2 + y'')^2} = 1+(y')^2\\ \frac{(y')^6 + 6 (y')^4 ...
1
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1answer
42 views

Proof that derivative of a function at a point is the slope of the tangent at the point

Why is the derivative for a function at point A considered the slope of the tangent of the function at this point?
9
votes
2answers
132 views

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$?

How to determine the existence of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, which is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$? It's more than enough to give an ...
2
votes
7answers
92 views

Prove that $ y''(0) = -1, x = \cos\left(\frac{t}{1+t}\right), y = \sin\left(\frac{t}{1+t}\right)$

My attempt: $$x'= -\sin\left(\frac{t}{1+t}\right) * \frac{1}{(1+t)^2}$$ $$y'= \cos\left(\frac{t}{1+t}\right) * \frac{1}{(1+t)^2}$$ then I attempted to divide y' on x' which resulted in ...
1
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1answer
48 views

$d(\beta \wedge d\beta)=0$ if $k$ is even.

Let $\beta$ be a $k$-form. Show that $d(\beta \wedge d\beta)=0$ if $k$ is even. I get that $d(\beta \wedge d\beta)=d\beta \wedge d \beta + (-1)^k\beta \wedge d^2\beta=d\beta \wedge d \beta$. Why ...
0
votes
1answer
28 views

Why is $ g'(s)=\frac{\partial f(t,y+s(x-y))}{\partial x}(x-y)$

If $g$ is defined as $g(s)=f(t,y+s(x-y))$ why is $\displaystyle g'(s)=\frac{\partial f(t,y+s(x-y))}{\partial x}(x-y)$ Does it not have to be $\displaystyle g'(s)=\frac{\partial ...
0
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0answers
44 views

N-th derivative of a expotential derivative

What is the $n^{th}$ derivative of function $$e^{- \large\left(1/{x^2}\right)}$$ Thanks.
1
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1answer
33 views

Finding intervals of increase/decrease after fundamental theorem?

$$F(x)=\int_3^x\frac9{\ln(2t)}\,dt\text{, for }x\ge3$$ So I'm aware $F'(x)$ is $\frac9{\ln(2x)}$. The question is asking what interval $F$ is increasing and which interval range it is concave up. I ...
0
votes
2answers
34 views

find the derivative with fundamental theorem of calculus

I came to cos(cos(sin(x)^3)+sin(x)), which is incorrect. I have a hunch that the error is somewhere around the sin x and the exponent. I'm just not sure what to do to fix it.
0
votes
2answers
41 views

$f(x) = \int_{0}^{x}(49-t^2)e^{t^3}dt$

If $\displaystyle f(x) = \int_{0}^{x}(49-t^2)e^{t^3}dt$, on what interval is $f$ increasing? How does one graph from the interval between a constant and an unknown variable? I don't understand. The ...
2
votes
1answer
30 views

Evaluate $f^{\prime}(0)$ where $f(x) = \begin{cases} g(x) \sin(1/x)& \mbox{if} \quad x \neq 0 \\ 0 &\mbox{if} \quad x = 0 \end{cases}$

I am trying to evaluate $f^\prime (0)$, where $$ f(x) = \begin{cases} g(x) \sin(1/x)& \mbox{if} \quad x \neq 0 \\ 0 &\mbox{if} \quad x = 0, \end{cases} $$ and $g(0) = g^{\prime}(0) = 0$. I ...
3
votes
0answers
60 views

If $f$ is differentiable at $x_0$, then $\lim_{h\rightarrow 0}(f(x_0 + ah) - f(x_0))/h = af'(x_0)$

If $f$ is differentiable at $x_0$ and $a\in \mathbb R$, show that $ \lim_{h\rightarrow 0} {f(x_0 + ah) - f(x_0) \over h} = af'(x_0)$ EDIT Suppose $u = ah $ then $ h = a/u$. So now we have $$ ...
0
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0answers
15 views

Help with a graphical derivative problem. [duplicate]

Let f(x) be a function such that f(0)=0, f(1)=1, f(2)=2, f(3)=4, and f'(x) is differentiable everywhere. Prove that there is a number a in the interval (0,3) such that 0< f''(a) <1.
1
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3answers
117 views

Continuity is required for differentiability?

My professor emphasized that: Differentiability implies continuity and Continuity is required for differentiability. Since a function like $\frac 1 x$ is differentiable but not continuous, I ...
2
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0answers
137 views

Project on slope, rates of change, and instantaneous rates of change

I was wondering if someone could look over my work and tell me if I am doing this correctly. Also, I need help on section D. Not understanding what values to substitute into the difference quotient. I ...
5
votes
1answer
112 views

Finding the derivative of $f(x) = \frac{8}{\sqrt{x -2}}$ using first principles.

How would you go about determining the derivative of ( $f(x) = \frac{8}{\sqrt{x -2}}$ ) using the limit definition of the derivative (i.e. $\lim\limits_{h\to 0} = \frac{f(x+h) - f(x)}{h}$) as opposed ...
0
votes
2answers
31 views

Rate of change problem with and without Leibniz notation

I'm trying to learn Leibniz notation, so I'm trying to understand how this problem could be solved with and without Leibniz notation so I might get a connection with what I know. I only know Lagrange ...
0
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0answers
20 views

Nomenclature — object of differentiation

We use the word integrand when integrating, e.g., $ \int x^2 \ dx = x^3/3 + c $, $ x^2$ is the integrand. When differentiating we tend to avoid using a word to denote the expression being ...
4
votes
3answers
114 views

General derivative of $\text{sinc}(x)$

I'm trying to find a general formula for the $n$-th derivative of the following function $$ \frac{\sin(x)}{x} $$ I've computed the first five derivatives so far without finding any 'structure' in it. ...
4
votes
1answer
92 views

Sequence of differentiable functions converging to non-differentiable function

Purely out of interest, I wanted to try and construct a sequence of differentiable functions converging to a non-differentiable function. I began with the first non-differentiable function that sprung ...
2
votes
2answers
57 views

Differentiability/continuity of piecewise defined functions

Let $$f(x)=\begin{cases}x^2\sin(\frac{1}{x}), &x\not= 0,\\ 0, &x = 0.\end{cases}$$ Since I can differentiate both parts of this, technically, $f$ is differentiable for all $x$. However I have ...
3
votes
3answers
46 views

How does this derivative notation work?

Elasticity of substitution = $\dfrac{\mathrm d \ln(k/l)\;\;\;}{\mathrm d \ln(f_l/f_k)}$ This is type of notation I haven't really worked with before. Is this read as "The change in the natural log of ...
1
vote
1answer
20 views

How to find where the function is decreasing/increasing/concave/convex $f(x) ={\frac{2}{1+x^2}}$?

$f(x) ={\frac{2}{1+x^2}}$ I need to find where this function is increasing, decreasing, concave and convex. I've found it's derivative: $f'(x)=\frac{-4x}{(1+x^2)^2}$ Now you're supposed to make ...
2
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0answers
33 views

How can we compute the integral of a Laplacian of a radial function over an open ball

Let $B_R\subseteq\mathbb{R}^n$ be an open ball with radius $R>0$ centered at $0$ and $f\in C^0\left(\overline{B_R}\to\mathbb{R}\right)$ be a radial function, i.e. $f(x)=f(r)$ with ...
1
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0answers
37 views

Show that for any neighborhood, derivative is not always positive

I need to show that for $\,{\rm f}\left(\, x\,\right)\equiv x + x^{2}\sin\left(\, 1/x\,\right)$ when $x$ is not zero and $\,{\rm f}\left(\, 0\,\right) = 0$, that there is no neighborhood of $x=0$ ...
0
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1answer
100 views

Value of Pi derivation

Derive the value of Pi. I want with explanation. Is there any possible way? How do scientists calculate it?
2
votes
1answer
55 views

derivative of a norm VS norm of a derivative

Consider a vector-valued function of the time, say $$v: \tau\in\mathbb{R}\to\mathbb{R}_N.$$ Suppose that for $\tau=t$, the function is equal to the zero vector, i.e. $$v(t)=0_N.$$ Denote as ...
3
votes
3answers
64 views

Explain why $|x^2-x|$ is not differentiable at $x=1$

Explain why $|x^2-x|$ is not differentiable at $x=1$..... lets go.... we need to show that $\lim_{a\to0}$ of $\frac{f(1+a)-f(1)}{a}$ doesn't exist...which is to say $\lim_{a\to0}$ of ...
0
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3answers
86 views

Prove the derivative of $\sin(1/x)$ exists

How do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...
0
votes
1answer
47 views

Is a constant a $ C^\infty $ function?

Is the function $f(x)= t $ (where t is a constant) a $ C^\infty $ function? The derivative would be zero, and it is continous everywhere. But still a classmate of mine doesn't seem to be convinced ...
5
votes
2answers
77 views

Rudin's chain rule: Why is continuity at interval necessary?

Theorem 5.5, Rudin's Principles of Mathematical analysis says: Suppose $f$ is continuous on $\color{red}{[a,b]}$,$ f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which ...
1
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1answer
27 views

What properties do I have if I know $f$ and $f^{-1}$inverse are differentiable?

My goal is to show that $(f^{-1})'(y) = 1/[f'(f^{-1}(y)]$ for all $y$ in $(a,b)$. I have no idea where to start. I know that $f^{-1}$ and $f$ are differentiable.
0
votes
2answers
48 views

complex differentiation, alternative way?

$\partial/\partial\bar{z}$ is defined as $1/2[\partial/\partial x+i\partial/\partial y]$. So lets say you have a function $f(z,\bar{z})$ in order to find $\partial f/\partial \bar{z}$ I have to write ...
1
vote
2answers
98 views

Laplacian of a radial function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a radial function, i.e. $f(x)=f(r)$ with $r:=\left\|x\right\|_2$. As stated at Wikipedia $$\Delta f=\frac{1}{r^{n-1}}\frac{d}{dr}(r^{n-1}f')$$ What's the most ...
0
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1answer
22 views

Differentiation of composed function.

Using the rule of differentiation composed function calculate the first order partial derivatives of $x$ and $y$: $$ z = f(u,v,w) = \arcsin \frac{u}{v+w}$$ $$u= e^\frac{x}{y}, v= x^2 + y^2, w =2xy$$ ...
4
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1answer
48 views

convolution with $C^{\infty}$ produces $C^{\infty}$

Problem: So I have the following function in ...
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votes
2answers
71 views

Why does differentiability implies continuity, but continuity does not implies differentiability?

Why does differentiability implies continuity, but continuity does not implies differentiability? I am more interested in the part about a continuous function not being differentiable. Well, all ...
0
votes
1answer
31 views

Limit of Derivative and Derivative of Limit

Assuming the integral is finite and $f$ is continuous, does this argument always work? If not, what do we need more? $\displaystyle \frac{d}{dx}\int_x^{\infty} f(t) \, \mathrm{d}t =$ $\displaystyle ...
0
votes
2answers
50 views

Condition for a line to be tangent of a parabola

I just read that the line y = kx + n will be tangent line of a parabola y^2 = 2px if derivatives of both of them are the same. ...
0
votes
3answers
60 views

Find the derivative of y = $\sqrt{xe^{2x} + 3e^{-x^2}}$

I am trying to find the derivative of this problem but I am not sure where to start. Any help is appreciated. Find the derivative of $$y = \sqrt{xe^{2x} + 3e^{-x^2}}$$
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1answer
36 views

Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
1
vote
1answer
52 views

Differentiability of monotonic functions

If a function is monotonic on set E. Is f differentiable almost everywhere? I have proved for case E closed bounded or open intervals, hence all open sets. But in general I am not able to figure it ...
0
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0answers
19 views

derivative of a specific integral

I would like to get the derivative of the function $f(z)$ defined as: $$ f(z)=\int_{z+dz}^{z}k\left(h(z')-a\right).g(z')dz' $$ where $a$ and $k$ are constant and $h(z)$ and $g(z)$ functions. How do I ...
6
votes
2answers
74 views

How to solve $\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$

Could you help me to prove $$\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$$ where ${\rm B}(a,b)$ is Beta function.
0
votes
1answer
72 views

Where is the slope of the tangent to an astroid equal to -1?

The equation of astroid is $x^{2/3} + y^{2/3} = a^{2/3}$. Find the points where the slope of the tangent to the astroid is equal to $-1$. I got the derivative to be $-y^{1/3}/x^{1/3}$ and so I ...
0
votes
2answers
36 views

Limit definition of derivative

How do I go about doing this question? Am i using the right formula?
1
vote
1answer
33 views

Can definite integral be of the form $\int_{h(x)}^{g(x)}F'(x)dx exist$?

In theory, I can write down an integral of the form $$I(x)=\int_{h(x)}^{g(x)}F'(x)dx$$ and solve it as $$I(x)=F(g(x))-F(h(x))$$ Out of curiosity, I plugged this into my calculator and was given a ...
1
vote
1answer
31 views

Derivative of integral with x as the lower limit

Question: Let $$F(x) =\int_{x^3}^{5}(cos^2t-te^t)dt $$ Find $F'(x)$ We were not explicitly taught about this during the semester but from what I can gather from online readings is that ...