# Tagged Questions

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### I need help on the process of solving this derivative.

How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$ I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
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### Derive an equation for derivative of ln x

$\frac{d}{dx}e^x = e^x$ use this fact together with the definition of the natural log $\ln x$ as the inverse of the function of $e^x$ to derive an equation for the derivative of $\ln x$.
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### Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } )$ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
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### Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
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### show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...
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### Logarithmic Differentiation

When do we use : $lnab = ln a + ln b$ and when do we use : $\ln |y| = \ln |f_1(x)| + \ln |f_2(x)| + \cdots + \ln |f_n(x)|$ ? It is stated that we use the second form of log differentiation ...
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### Derivative of $\operatorname{Log}(\operatorname{Log}(z^2))$

Please help me with this question: (i don't know how to start) Suppose that $f(z)$ = $\operatorname{Log}(\operatorname{Log}(z^2))$. Find $f'(z)$ where it exists, and determine the set of points at ...
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### Finding $y^{\prime}$ of $y=\log_7 e^{8x}$

Find $y^{\prime}$ of $y=\log_7 e^{8x}$ I know that $\dfrac{d}{dx}(e^{8x})=8e^{8x}$, but I am confused on how to work the rest of the problem. Is this correct: $\log_ex=\ln x$ and that ...
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### Help with logarithmic differentiation problems

$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ My try: $\ln y=\ln(8)^{\sqrt x}$ $\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$ I don't know how to proceed with right side. $\mathbf{(2)}$ Find ...
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### How to get the derivative of $(\ln(x))^{\sec(x)}$?

How do you get the derivative of $(\ln(x))^{\sec(x)}$? I know that the derivative of $\ln(x)$ is $\frac 1x$ but what happens when you take it to an exponent of $\sec(x)$?
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### Hessian of a conic function

i got a conic System: $Ax =b, x\in C$, where $A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m$ and C is the cone of the $n\times n$ positive semidefinite matrices, so ...
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### Use a graph to estimate the time at which the number was increasing most rapidly

For the period from 2000 to 2008, the percentage of households in a certain country with at least one DVD player has been modeled by the function $f(t) = \frac{87.5}{1 + 17.1e^{âˆ’0.91t}}$ where the ...
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### Properties of Natural Logarithm I need help finding the Derivative

$y=\ln(x)^2$ I am not sure why the answer would be $\frac{2\ln(x)}{x}$ I used this property "power rule" "$\ln(x^n) = n\ln(x)$ So i got $2\ln(x)$ the derivative of that using the constant ...
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### Deriviative of natural log help finding

$$y=7\ln\frac{11}x$$ I need to use the product rule please $$\frac{d}{dx} (7) \cdot(\ln(11/x) + \frac{d}{dx}\left(\ln\frac{11}x\right) \cdot 7$$ then $$0+\frac{d}{dx}\frac{11}x \cdot7$$ what do ...
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### Derivative of f(x)=arth(lnx)

I'm struggling with finding derivative of f(x)=arth(lnx). I've done following: x=th(y) ...
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### How to evaluate derivative in the form of $\log^n_nx$

How does one evaluate this derivative? $$\frac d{dx}\left[\sum^N_{n=2}\log^n_nx\right]$$ Work so far: The differentiation operator and the sigma can be easily swapped but it's more of a question ...
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### Closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$

I need a closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$. Here $A, X$ are $n\times n$ matrices, $\log$ is the matrix logarithm. ${d \operatorname{Tr}(\log(X))\over dX}=X^{-1}$, ...
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### Help with differentiation of natural logarithm

Find $\;\dfrac{dy}{dx}\;$ given $y=\frac{\ln(8x)}{8x}$. The answer is $\;\dfrac{1-\ln(8x)}{8x^2}\;$. Can you show the process of how this is worked? Thanks.
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### Looking for help understanding the asymptotic expansion of the digamma function

I was recently given an example using this asymptotic expansion of the digamma function where: $$\frac{d}{dx}(\ln\Gamma(x)) = \psi(x) \sim \ln(x) - \frac{1}{2x} - \frac{1}{12x^2}$$ Here's the ...
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### Derivative for log

I have the following problem: $$\log \bigg( \frac{x+3}{4-x} \bigg)$$ I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points ...
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### Derivative of compositum function with log

I have the following two functions that I'm not compleately sure I'm solving correctly mainly what bugs me is $\log(x)$. 1st Function: $$f(x) = \sin(2x^2 - 3\log(x))$$ I simply treated this as ...
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### Differentiation Of Natural Logarithms

The problem I have is to differentiate $y = ln(x^4)$ Using the rule : $$\frac{d[lnf(x)]}{dx}=\frac{f'(x)}{f(x)}$$ My working is: $$\frac{dy}{dx} = \frac{x^4ln(x)}{x^4}$$ $$=ln(x)$$ but the book is ...
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### Arithmetically showing that $\frac{\log(x+1)}{\log(x)}<\frac{x+1}{x}$

Is there a possibility that this can be shown arithmetically? By arithmetically, I mean not looking at the graph. $$\frac{\log(x+1)}{\log(x)} < \frac{x+1}{x}$$ Thank You
It is well known that $$\int \frac{f'(x)dx}{f(x)}= \int d \log f(x)=\log f(x) + C$$ In my work I came across the following case: $$\int \frac{(f'(x))^2dx}{f(x)}$$ I wonder if any interesting ...
When applying the definition of a derivative to $\frac{d}{dx}b^x$ and a little algebra one arrives to $$b^x\times\lim\limits_{h \to 0}\frac{b^h - 1}{h}$$ where of course that limit equals $\ln(b)$. ...