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

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1answer
57 views

Where am i going wrong in solving this equation?

Fing the least value of $a$ for which $f(x)$ is increasing, where $$f(x)=2e^x-ae^{-x}+(2a+1)x-3$$ What i tried for increasing $f'(x)\ge 0, \forall x\in \mathbb R$. So ...
0
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1answer
12 views

Rate of change for a volume

I have the following question : The radius of a right circular cone is increasing at a rate of 5 inches per second and its height is decreasing at a rate of 4 inches per second. At what rate is the ...
1
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2answers
58 views

If $f$ is differentiable in $(a,b)$ then $\frac{1}{f}$ is differentiable at $(a,b)$, provided $f(a,b)\neq0$

"Suppose that $f$ is a differentiable function at $(a,b)$. Prove that $\frac{1}{f}$ is differentiable in $(a,b)$, provided $f(a,b)\neq0$" We were given the following definition of differentiability: ...
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2answers
61 views

Evaluating the inverse trigonometric limit $\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}}$

$$ \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}} $$ I was doing some questions on limits, I saw one in which there is $\arccos x$. I am stuck ...
5
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1answer
69 views

Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...
2
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2answers
32 views

Infinite differentiability of a function with a removable discontinuity

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely ...
1
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1answer
38 views

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
2
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3answers
88 views

Applying the chain rule to compute $\frac{d}{dx}(\cos^6 x)$

$$\frac{d}{dx}(\cos^6x)$$ Using the chain rule $ M'(N(x)).N'(x)$, I'm deconstructing the $\cos$ function $$\begin{align*} &M= \cos^6 \\ &N= x\end{align*}$$ End result should be ...
0
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0answers
23 views

Why is the following derivation correct?

Say we have f=f(x,t) x=x(x',t') Wangsness (2nd ed., chap. 29) does the following procedure to find the partial derivatives: $$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial ...
0
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0answers
7 views

Testing differentiability and continuity

Consider the following function $ f(x) = 0 $ if x is rational $ f(x) = x^2$ if x is irrational Then only one of the following statements is true which one is it ? a.) $f$ is differentiable at ...
5
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2answers
82 views

Derivation of Dirac delta function

Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
-1
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1answer
29 views

Find the derivative and integral of the following function

I'm a bit confused on how to work out this question, so if you could show working it would be much appreciated. Thanks. Find $f'(x)$ and $\int f(x)\,dx$ for $$f(x)= ...
2
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1answer
81 views

Pushforward of a vector

The push forward of vectors allows to transform the components of a vector $X$ to be pushed along a map $h:\mathcal{M}\to\mathcal{N}$ between the manifolds $\mathcal{M}$ and $\mathcal{N}$. This is ...
1
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1answer
52 views

Find the derivative of $\left(\frac{4x+2}{x-2}\right)^5$

Hey helpful people I have one more question I am stuck on! $$f(x) = \left(\frac{4x+2}{x-2}\right)^5$$ I know the answer is $$\frac{-50(4x+2)^4}{(x-2)^2(x-2)^4}$$ But I really can't figure out how ...
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0answers
8 views

Inequality with derivative and supremum norm

I have the following property written in a book but I can't understand why this implication is true. I would be glad if anyone could help me let $A \in \mathbb{R}^N$. $$\frac{d}{dt} \nabla A(x,t) = ...
3
votes
2answers
211 views
+50

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
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0answers
25 views

Differentiation of an integral in regards to different variables

It is known by the second fundamental that $$\frac{d}{dx}\int_0^x{\sin{(a \cdot t)}\ dt}=\sin{(a \cdot x)}$$ But what can we say about $$\frac{d}{da}\int_0^x{\sin{(a \cdot t)}\ dt}=\ ?$$
1
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1answer
501 views

Gradient function of a circle

The parametric equations of a circle $C$ are: \begin{align*} x&=2+\dfrac{13}{5\sqrt{2}}\cos t\\ y&=1+\dfrac{13}{5\sqrt{2}}\sin t \end{align*} for $t\in[0,2\pi]$. I am stuck on this part: Find ...
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6answers
2k views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
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3answers
89 views

Derivative of $\log |AA^T|$ with respect to $A$.

What is the derivative of $\log |AA^T|$ with respect to $A$ where $|A|$ denotes the determinant of matrix A?
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2answers
61 views

If $f\in C^1[a,b]$, then it can be expressed by the sum of an increasing function and a decreasing function.

Prove: If $f$ is a continuous class $1$ function on $[a,b]$ then it can be expressed by the sum of an increasing function and a decreasing function. I don´t know where to start my demonstration, I ...
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3answers
49 views

Find the derivative of $\frac{3}{x} - \frac{x}{2}$

I must find the derivative for: $\frac{3}{x} - \frac{x}{2}$ I know the answer is$ \frac{-3}{x^2} - \frac{1}{2}$ But I can't figure out why the 3 is negative and where the 1/2 came from Any help ...
3
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1answer
44 views

Finding the derivative of $f(x) = 2x^2 + x - 3$ at $x = 4$.

I am learning about derivatives and differentiating and I came across this; $f(x) = 2x^2 + x - 3$ at $x = 4$ This is as far as I get; $$\frac{2(x + h)^2 + (x + h) - 3 - (2x^2 - x + 3)}{ h }$$ ...
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0answers
29 views

How does the step in the picture transition to step 2?

:) I have a math question regarding this picture. The problem is that I do not understand how the first equation turns into the the second. Where did the integral come from?? (the dv and dt) Update: ...
2
votes
2answers
74 views

The notion of “infinitely differentiable”

Wiki takes me to the section "smoothness" which I don't entirely get, it's just too much stuff for me. My question is, what exactly is it? An infinitely differentiable function is one that is ...
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3answers
30 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
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0answers
30 views

$f$ is $3-$times differentiable and has at least $5$ distinct real zeroes, prove $f+6f'+12f''+8f'''$ has at least two distinct real zeroes?

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeroes. Prove that $f+6f'+12f''+8f'''$ has at least two ...
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0answers
21 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: ...
9
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3answers
247 views

100th derivative of $e^{-x^2}$ at point $0$

Problem: Find $\frac{\mathrm d^{100}}{\mathrm dx^{100}}e^{-x^2}$ at point $0$. My attempt: $y'=-2xe^{-x^2}$ I tried to use General Leibniz rule and I didn't get much better information. Without: ...
0
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1answer
37 views

$n$th derivative of function $\frac{1}{(1-2x)^2}$

I am trying to find the $n$th derivative of the function $\frac{1}{(1-2x)^2}$. The first three are simple but I can't see a schema right now. \begin{align*} y^{\prime} & = \frac{4}{(1-2x)^3}\\ ...
0
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1answer
23 views

What is the full width of a peak of the function $F(X)=\frac{1+\cos((2N+1)πX)}{1+\cos(πX)}$

With $$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ It peaks at odd X integer values. ...
2
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2answers
46 views

How to find the set of values $S$ where $f$ is not differentiable?

Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no ...
4
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1answer
30 views

Derivative of x|x| at 0

I am trying to show that $f(x) = x|x|$ is differentiable for all $x \in \mathbb{R}$. By computing the prime derivative I get: $$f'(x) = |x|+x(|x|)'$$ I know that $(|x|)' = \begin{cases} 1 \ ...
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1answer
49 views
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Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ ...
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2answers
83 views

Formula for the nth Derivative of a Differential Equation

I have the differential equation $$f'(x)=2xf(x)$$ With the initial condition that $f(0)=1$ I need to prove that the nth derivative evaluated at zero is equivalent to $n!/(n/2)!$ for even n. ...
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3answers
99 views

Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow ...
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2answers
44 views

Question on Rolle's theorem involving roots

Use Rolle's theorem to show that $f(x)=x^3-\frac{3}{2}x^2+\lambda$, $\lambda \in \mathbf{R}$ never has 2 zeroes in $[0,1]$. I started by assuming that $\exists$ $2$ zeroes in$[0,1]$ Then ...
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1answer
23 views

Determine the Fourier series considering the derivative of a function

Let $f\left(x\right)=x^2+1$ on the interval $\left[-\pi,\pi\right]$, which is extended periodically to $\mathbb{R}$. I have calculated the Fourier series of $f$ to be ...
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4answers
647 views

Continuity of Derivative at a point.

Is it possible that derivative of a function exists at a point but derivative does not exist in neighbourhood of that point. If this happens then how is it possible. I feel that if derivative exists ...
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1answer
44 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
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1answer
78 views
+50

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
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0answers
31 views

Problem finding the tangent plane and the normal line of an surface [on hold]

Good night, I have a serious problem when I try to find a tangent plane for the following surface at the point $P$: $$x^{2}+y^{2}+z^{2}=6, \hspace{4mm} P=(-1,-2,3).$$ I make this: $\nabla ...
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4answers
673 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
0
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0answers
17 views

Factoring $R(ry')'-y(rR')'=[r(Ry'-R'y)]'$

In a problem this formula was used and I'm not seeing how this factor using the chain rule was derived. Other than calculating the derivative of the two that someone else already solved and showing ...
3
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2answers
27 views

Relative Extrema - First-derivative test of : $f(x)=x^5-5x^3-20x-2$

Find the relative extrema of the function by applying the first-derivative test: $$f(x)=x^5-5x^3-20x-2$$ So I found the $f'(x)$ $$f'(x) = 5x^4-15x^2-20$$ Now, I'm trying to find the critical ...
2
votes
2answers
37 views

Partial Derivative of $xy^2+yz^2+xyz+x^2y^2z^2=5$

Someone can tell me what the Partial Derivative of $\frac{d^2z}{dy^2}$ of function $z(x,y)$ if it`s look like this: $$xy^2+yz^2+xyz+x^2y^2z^2=5$$ I try to solve the first derivative: ...
3
votes
2answers
62 views

When is $\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$ a valid approximation?

It is often said that when the change in e.g. $\Delta x$ is small than we can make the approximation: $$\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$$ But it is not enough to say $\Delta x$ is small ...
0
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1answer
41 views

If $f\colon\mathbb{R}\to\mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2$ for every $x\in\mathbb{R}$, then $f$ is differentiable at 0.

If $ f\colon \mathbb{R} \to \mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2 $ for every $x \in \mathbb{R} $, then $f$ is differentiable at $0$. The solution provided uses delta-epsilon to prove ...
1
vote
1answer
45 views
+50

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
0
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1answer
41 views

How the derivatives are different if sign changes.

I have this expression $$\frac{1}{(1 - x) ^ 2}$$ I need the derivative of this expression. So I calculated it, no big deal. However something has crossed my mind. Mathematically $(1 - x) ^ 2 = (x - 1) ...