All Questions
Tagged with logarithms derivatives
450
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When Does ((n^a)-1)/a)) Equal e; A Sophomore's plight
I am a high school student (sophomore) and have come across something I would like explained.
I was watching 3blue1brown for an explanation of calculus, when he used the formula: lim a->0 (d/dx(n^x)...
3
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4
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Derivative of $e^{x+e^{x+e^{x+...}}}$
Let $y=$ $e^{x+e^{x+e^{x+...}}}$
To find $\frac{dy}{dx}$, I took the natural log on both sides, which gives: $$\ln y = x + e^{x+e^{x+e^{x+...}}}$$
Differentiating on both sides,$$\frac{1}{y}\frac{dy}{...
- 127
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Why does the scalar inside a natural log dissapear when differentiating it? [closed]
For example if I was differentiating $\ln(2x)$ doesn't the chain rule dictate that it should be $2/x$, not $1/x$? Why does the $2$ disappear?
user1329181
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How does $\frac1{f(x)}\cdot\frac d{dx}f(x) = \frac d{dx}[\ln f(x)]$?
Perhaps a basic question, as I was told that this was easily solvable using the chain rule, although I can't figure out why:
$\frac1{f(x)}\cdot\frac d{dx}f(x) = \frac d{dx}[\ln f(x)]$
I am wondering ...
- 204
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2
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Formula for higher-order derivatives of the logarithm of a function, $\frac {d^m}{dx^m} \log h(x)$
I am trying to see if I can get a general formula for the $m$th derivative of the logarithm of a function. Specifically, I am trying to find the $m$th derivative of $g(x) = \log h(x)$.
Let $h^{(m)}(x) ...
- 317
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does Integrating both sides of an equation in dx will Invalidates the equality?
I'm struggling to grasp the justification behind integrating both sides of an equation. While I understand that operations can be applied to both sides, maintaining equality, it appears that this ...
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I was trying to understand QUAKE III fast inverse square root alg and i want to find best 'u' value in $\log_2(x+1)≈x+u$ approximation
I was trying to find best 'u' value for this approximation:
$\log_2(x+1)≈x+u$
And I did think I can calculate error with this function.
NOTE: for the x values between 0 and 1 i need becouse of IEEE ...
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1
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Evaluation of $\int_{0}^{\infty} \, \frac{x^\alpha \log^n{(x^\beta)}}{k^m+x^m}\, dx$
How to evaluate the following integral/
$$\int_{0}^{\infty} \, \frac{x^\alpha \log^n{(x^\beta)}}{k^m+x^m}\, dx $$, $\alpha, \beta, m, n$ are real and $k$ is a positive integer.
For the case where $\...
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Simplify $n$th derivative of $\frac{\log^3 x}{1-x}$
I need to simplify $n$th derivative of $$f(x)=\frac{\log^3 x}{1-x}$$ where $0<x<1$
I tried writing $f(x)=u_1u_2u_3u_4$ where $u_1=u_2=u_3=\log x$ and $u_4=\frac{1}{1-x}$.
Using Leibniz rule for ...
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1
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Approximating $(\log n)^n$ using Big O notation
As stated in the title I am trying to find a good approximation in O notation for:
$$(\log n)^n = \log (n) \cdot \log (n) \cdot ... \cdot \log (n)$$.
I have tried using L'Hôpital's rule for the term
$...
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105
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Evaluate: $\int_{-2}^{2} \frac{x^2}{1+5^{x}} dx$
Evaluate:
$$\int_{-2}^{2} \frac{x^2}{1+5^{x}} dx$$
After checking f(-x) for odd/even function and not getting suitable results.
Let $$1+5^x=t$$
$$dx=\frac{dt}{log(5)(t-1)}$$
$$x^2=\frac{(t-1)^2}{(log(...
- 155
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How to compare two logarithmic functions using derivatives? [closed]
I have two expressions: $\log_2 5$ and $\log_3 13$.
And I need to determine which expression is greater using exactly derivatives and not having a calculator.
I know that $(y = \log_2 5)'$ = $\frac{1}{...
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How to prove $x^{\ln x} > \frac{x}{2} + \frac{1}{2x} $?
How to prove the following? $$x^{\ln x} > \dfrac{x}{2} + \dfrac{1}{2x} \tag{1} $$ for all $x \in \mathbb{R}^+\setminus \{1\}$?
I could prove $ x^{\ln x} > x $ and
$ x^{\ln x} > x/2 $ (in ...
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Finding maxima of a log and a linear term
I am supposed to study the function $$f(x)=x\cdot \ln(x+2)$$ To find the maxima and minima, I am supposed to set its first derivative $$f'(x)=\ln(x+2) + \frac{x}{x+2} \ge 0$$
However, I am unsure of ...
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Gradient of a $\log\det$ function
Given symmetric $d \times d$ matrices ${\bf G}, {\bf F}_1, \dots, {\bf F}_n$, let ${\bf F} : {\Bbb R}^n \to {\Bbb R}^{d \times d}$ be defined by
$$ {\bf F} ({\bf x}) := {\bf G} + \sum_{i=1}^n x_i {\...
