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

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1answer
35 views

Differentiation of a modulus function

How to find derivative of $$f(x)=|\sin^{-1}(2x^2-1)|$$ Please provide stepwise mechanism. The original question was to find domain of derivative of y=|arc sin(2x^2−1)|. My METHOD- My attempt was ...
0
votes
1answer
29 views

Differentiating an indirect function

Question: Find $\frac{dy}{dx}$ if: $$x^2 + y^2 = t + \frac{1}{t}$$ and $$x^4 + y^4 = t^2 + \frac{1}{t^2}$$ Attempt: To find $dy \over dx$, we basically need to find $dy \over dt$ and $dx \over ...
2
votes
1answer
62 views

How to solve $x=\sin^{-1}(\frac{1}{2\sqrt{x}})$

I was setting a question when I came across a problem. The question was: Suppose I have a function $y=e^{1+\cos(x)+\sqrt{x}}$. (A) Locate its turning points by taking derivatives and sketch its graph ...
-2
votes
0answers
26 views

derivative of an equation containing min

I would appriciate if someone could help with letting me know how to take the dervivative of $\large{\frac{dDt}{dg}}$ of the following equation: $${Dt ...
0
votes
1answer
55 views

Visual difference between limits and derivatives?

I've heard of limit, continuity and derivative explained algebraically but I was wondering about visually too. I was wondering if someone could explain the purpose/difference of limit, continuity and ...
-1
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2answers
67 views

Is there a rigourous proof about this :$(C)'=0$ , where C is any constant?

Is there someone who can give me a rigourous proof about the derivative of any constant $C$ is zero :$(C)'=0$ Note :I know only this for any real number $x>0$:$(C=Cx^0)'=C'=0Cx^{0-0}=0.1=0$ ...
1
vote
3answers
51 views

Understanding total derivative

Given the information that $\vec A= \vec A(\vec r(t),t)$, why is $$ \frac{dA}{dt} = \frac{\partial A}{\partial t} + (\vec r' \nabla)\vec A$$ and not $$ \frac{dA}{dt} = \frac{\partial A}{\partial t} ...
3
votes
2answers
55 views

Does there exist higher degree graded derivations on $\Omega(M) $

Does there exist any other graded derivation on $\Omega(M)$ other than the one of degree one which is the exterior derivative (i.e. maps such as $d: \Omega^p(M) \rightarrow \Omega^{(p+r)}(M) $, where ...
0
votes
0answers
37 views

What will be Terms after repeating this step(Differentiation and multiplication) F times.

I was solving a probability problem and got stuck on the following situation, where each x_i is independent of others: $$f=(x_1+x_2+..x_k)^N$$ I'm interested in the expression obtained after ...
0
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2answers
37 views

Other forms for the derivative of the Gamma function

When I searched for the derivative of the Gamma function I got something of the form: $$\Gamma'(x)=\Gamma(x) \psi(x)$$ But from the definition of the Digamma function to me it's like writing: ...
1
vote
1answer
51 views

Infinite derivative of nested radicals

$$\cdots \frac{d}{dx}\frac{d}{dx}\frac{d}{dx} \sqrt{x+\sqrt[3]{x+\sqrt[4]{x\cdots}}}$$ Its not super hard to find a finite number of derivatives, but I can not understand how to pull off infinite ...
3
votes
0answers
27 views

Is there a differentiable function on a closed subset of $\mathbb{R}^n$ that cannot be continued differentiably on an open superset?

Let $A \subseteq \mathbb{R}^n$ be closed with no isolated points and $f:A \to \mathbb{R}^m$. Suppose that for every point $x_0 \in A$ we have (at least one) matrix $L_{x_0}$ such that $$ \lim_{x,y \to ...
0
votes
0answers
26 views

find local inverse functions of $f(x,y)=(x\sin(y),x\cos(y)), (x,y)\in (0,\infty)\times (0,3\pi)$

I have to determine the local inverse functions of $f(x,y)=(x\sin(y),x\cos(y)), (x,y)\in (0,\infty)\times (0,3\pi)$. I first proved that the determinant of the Jacobian matrix is nonzero for all ...
3
votes
1answer
105 views

Computing the volume of this weird object,

Let $f: [-1,1] \to \mathbb{R}$ be a continuously differentiable function such that $f(-1) = f(1) = 0$ and $0<f(x)\le 1$ for all $x \in (-1,1)$. Let $S$ be the surface in $\mathbb{R}^3$ obtained by ...
3
votes
1answer
48 views

Frechet derivative of squared norm $\|x\|^2$

I got this from Analysis II, H. Amann, J. Escher, p. 152. Their definition of the derivative of a map $f$ between Banach spaces $E,F$ over the field $\mathbb{K}$ is a bounded linear operator ...
1
vote
1answer
42 views

Tangent plane and tangent lines to curves through a point

Let $S$ be the surface that is the graph of a continuous function $f: U \rightarrow \mathbb{R}$ on an open $U \subset \mathbb{R}^2$. Let $p = (x, y, f(x, y)) \in S$. One usually defines the tangent ...
0
votes
1answer
38 views

Why is this expression evidently differentiable?

I came across this expression while reading a journal: $\hat{m}_{h}(\tau)=\frac{{\displaystyle {\scriptstyle {\displaystyle \sum_{s=t}^{t+l+d-1}}K_{h}(\tau-s)P_{s}}}}{{\displaystyle ...
4
votes
2answers
34 views

Basic question about nonstandard derivative

I'm trying to understand how the nonstandard derivative works. For instance, consider the function $f(x) = \frac{1}{2} x^2$ The derivative is $f'(x) = st \left( \frac{\frac{1}{2}(x + \epsilon)^2 - ...
1
vote
2answers
28 views

Maxima of a function

Question: If $$f(x) = \cos\frac{\pi x}{2015}; x>0$$ and $$f(x) = 2x + a; x\leq 0$$ Find the values of $a$ such that $x = 0$ is a point of local maxima for $f(x)$ Attempt: As it's a ...
1
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1answer
33 views

Differentiate the Function: $y=\sqrt{1+2{e^{3x}}}$

$y=\sqrt{1+2{e^{3x}}}$ I am using this formula $(e^{f(x)})'=e^{f(x)}\cdot f'(x)$ The Chain Rule $\begin{align*} f'(x) = g'(h(x)) h'(x). \end{align*}$ and the product rule $k(x)=f(x)g(x)\ then\ ...
0
votes
2answers
34 views

Solve linear differential equation

So I have the following linear differential equation $$t\frac{dy}{dt}-3y=t^4$$ My first step was to divide through by $t$ to give $$\frac{dy}{dt}-3t^{-1}y=t^3$$ Then to find the integrating factor ...
0
votes
2answers
56 views

How to prove this limit of derivative

Here is a question that I need help to prove it. Let $f:\mathbb{R}\to\mathbb{R} \in C^{\infty}$ be periodic of period $1$ and nonnegative. Show that $$ ...
1
vote
3answers
57 views

Differentiate the Function: $y=e^{k\ tan\sqrt{x}}$

$y=e^{k\tan\sqrt{x}}$ $=e^{k\tan\sqrt{x}}\cdot [{k\tan\sqrt{x}}]'$ $=e^{k\tan\sqrt{x}}\cdot\ (k)\cdot[\tan x^{\frac{1}{2}}]'+(\tan x^{\frac{1}{2}})\cdot[k]'$ $=e^{k\tan\sqrt{x}}\cdot\ (k)\cdot ...
1
vote
1answer
44 views

Why are higher order derivatives linked to the higher orders of convergence?

My textbooks defined the rider of convergence as follows (original image link) For an iterative process of the form $x_{n+1} = g(x_n)$, the order of convergence is first order when $|g'(x)| ...
1
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2answers
44 views

What is the derivative of $\log_x(A)$ where $x$ is the base (differentiaition with respect to $x$)

I want to find out what $\frac{d}{dx}\log_x A$ is? I did this so far but I'm not sure. $y = \log_x A \Longrightarrow x^y = A$ so, $d/dx(x^y) = d/dx(A)$ [differentiating both sides w.r.t $x$] then, ...
0
votes
2answers
48 views

Multivariable Functions - Second Derivative Problem

So here is the problem: Calculate the second class derivative on $(1,1)$ of the equation $x^4+y^4=2$ I found this problem on my proffesor's notes. However it doesn't state whether a partial or a ...
1
vote
4answers
62 views

Differentiate: $y=e^{ax^3}$

$y=e^{ax^3}$ What in a problem indicates I should use the laws of logarithm? Here is how I differentiated the function: $\ln(y)=\ln{e^{ax^3}}$ $\ln(y)=ax^3\ln(e)$ ...
0
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1answer
7 views

Gradient in cylindrical coordinates

This is more of a maths question, but several sources point at different expressions for the gradient in cylindrical coordiantes. Sometimes I see the radial component for the gradient of a scalar ...
-1
votes
3answers
134 views

How to evaluate $y'$? [closed]

$$yx^{2} = x^{y^{y}}$$ Question: $y'=?$
4
votes
4answers
222 views

Derivative of a definite integral issue

$g:\mathbb{(0,1]}\to \mathbb{R}$ We have the function $$g\left(x\right)=\int _x^1\left(\frac{\sin\left(t\right)}{t}dt\right)\:$$ Show that the function is strictly decreasing. So I thought that I'd ...
0
votes
5answers
61 views

Differentiate the Function: $f(x)=(x^3+2x)e^x$

$f(x)=(x^3+2x)e^x$ by looking at it I would assume I should use the product rule. Thus, $f'=x^3+2x\cdot e^x+e^x\cdot (3x)(2)$ $f'=x^3+2x\cdot e^x +e^x\cdot 6x$ However, this is not the answer ...
0
votes
0answers
7 views

How to show monotonicity of this weighted average?

I have a sequence of positive and increasing (in all of its arguments) functions , $\{f_i(x_1, x_2, \dots, x_M)\}_{i=1}^M$. I also have: $$ ...
2
votes
1answer
49 views

Partial derivative with respect to intermediate variable

Suppose we have $f(x,y,z)$, where $x=g(r,\theta,\phi)$, $y=h(r,\theta,\phi)$, $z=t(r,\theta,\phi)$. How do we find partial derivative with respect to $x$ and express it as a "function" of $r$, ...
3
votes
2answers
94 views

Find an equivalent of this function,

a) $f$ continuous on $[0,1]$ such that $f(x)>0$. Find an equivalent of $$h(\epsilon) = \int_0^1 \frac {f(x)}{x^2 + \epsilon^2}dx$$ when $\epsilon$ goes to zero and when $\epsilon$ goes to ...
2
votes
1answer
27 views

Proof of Liouville's formula , details and confusions. [Matrices, determinants..]

So I've got the homogeneous linear equation: $$x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ where $a_1(t)...a_n(t)$ are real continuous on intervals. This is what my textbook states: If ...
2
votes
1answer
29 views

Calculating the Lie algebra representation of the regular representation on subspace of functions on $\mathbb R$.

Let $G = \mathbb R$ and let $\pi$ be the regular representation of $G$ on $L^2(\mathbb R)$, that is, $\pi(g)(f)(x) = f(x-g)$ for $g \in G$. Let $V = \{f \in \mathcal C_c^\infty | supp f \subseteq ...
0
votes
1answer
63 views

Is there a function that doesn't have a limit at infinity but its derivative does?

Is there a function that is differentiable in $\mathbb{R}$, its derivative converges to $0$ as $x\to \infty$ but the function does not converge as $x \to \infty$, neither to a finite limit nor to an ...
0
votes
1answer
12 views

Derivatives expression understanding problem

From book about hydraulics I saw while reading: $$ \frac{du_x}{dt} = \frac{\partial u_x}{\partial t} + \frac{\partial u_x}{\partial x} u_x +\frac{\partial u_x}{\partial y} u_y +\frac{\partial ...
1
vote
2answers
49 views

Differentiation of complex valued functions. $i^x$

The question is how to find the derivative of $i^x$ or even if it exists?. WolframAlpha does give an answer. Before applying the definition, we see that how can we mix the complex and the real plane. ...
0
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1answer
42 views

What's meaning of a derivative to its original funciton?

I have trouble of finding the relationship of a function's derivative with its original function. Suppose there is a function: $$s = f(x)=16x^2 + 2$$ So its derivative is $$s'=f'(x)=32x$$ Here's ...
2
votes
1answer
74 views

Is a function a derivative?

I'm reading introductory calculus and I find that 'function' tends to be defined by what it does rather than what it is. If $y = f(x)$, then surely the value of $y$ is dependent on that of $x$, i.e. ...
2
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2answers
61 views

Integrating a First Order Differential Equation (The West Equation)

I am currently doing a project about Growth and have found this really interesting Math Model by Dr. Geoffrey West et al in 2001 while researching. The paper can be found at this link. I was ...
0
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1answer
31 views

Is there a concept called the cross derivative between two functions?

Let $f$ and $g$ be two real functions. Is there already a concept for the quantity $\lim_{h \to 0} \frac{f(x+h)-g(x)}{h}$? Note that when $g=f$, the quantity, if exists, is the derivative of $f$ at ...
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vote
2answers
62 views

Find $y'$ if $y=e^{-4x} \sin\ 5x$

$$y=e^{-4x} \sin\ 5x$$ My answer is: $y' = e^{-4x}(\cos\ 5x)(5)+(\sin\ 5x)e^{-4x}$ = $$e^{-4x}((\cos\ 5x)(5)+(\sin\ 5x))$$ The books answer is different am I right?
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8answers
149 views

Differentiate the Function: $y=e^{\tan x}$

$y=e^{\tan x}$ The book says to use the Chain Rule. Let $u = \tan x$. Thus, $y = e^u$ $du = \sec^2x\ dx$ $\frac{du}{\sec^2x}= dx$ I am confused at this point. The book explains the method of ...
1
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2answers
31 views

Having 2 functions of the same variable, how can I find the derivative of the first function in relation to the other?

Let's be specific and use a simpler example than what I actually need to solve. $$ \begin{split} x(t) &= t + A\sin(wt) \\ y(t) &= B \cos(wt) \end{split} $$ How would I obtain the derivative of ...
3
votes
2answers
320 views

Can the following trick be expanded upon?

Main Question What is the expansion of $d^{1+\epsilon}?$ Background I noticed the following trick (sometimes more laborious) to directly differentiate $ f(x) $ twice without differentiating it even ...
3
votes
2answers
87 views

Show $ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$ using Taylor

Let $f:[a, b]\to R$ differentiable at $a<x_0<b$. Using taylor series show that if $x_n \to x_0^-$ and $y_n \to x_0^+$ then $$ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$$ ...
1
vote
1answer
39 views

Derivative of integral and variable substitution

My question is about the validity of this identity and if there is some error in my argument: $$\int_0^{\infty}\frac{d}{dt}f(t-x)dx = -\int_0^{\infty}\frac{d}{dx}f(t-x)dx$$ The argument goes as ...
7
votes
1answer
60 views

Semantics of Writing Differential Equations

Let $f : \mathbb R \to \mathbb R$, and consider the differential equation $$ f'(t) = f(t) $$ it is easily seen that it has the solutions $f(t) = a\cdot \exp(t)$ for $a \in \mathbb R$. Now another way ...