1
vote
4answers
48 views

Unable to differentiate $\cos(x) \cos(2x) \cos(3x)$ and $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

I apologize for the lack of LaTeX. I will update this question with the proper LaTeX as soon as possible. I am having trouble with two differentiation exercise questions and was hoping someone could ...
3
votes
1answer
62 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
0
votes
0answers
13 views

Exponential and Logarithmic Differentiation.

Q. If $xe^{xy}=y+sin^2x$, then find $\frac{dy}{dx}$ at x=0. If we differentiate the function directly as follows: $e^{xy}+xe^{xy}\left[y+x\frac{dy}{dx}\right]=\frac{dy}{dx}+sin\left(2x\right)$ At ...
0
votes
0answers
18 views

Linearizing non-linear least squares: Problem with derivatives

We want to approximate $$y_i \approx a b^{x_i}$$ and thus have $$S=\sum_{i=1}^m (ab^{x_i}-y_i)^2$$ as least squares error term. This term is not linear in b, so it is not easy to calculate its ...
2
votes
2answers
55 views

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 ...
2
votes
1answer
74 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
0
votes
2answers
43 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
0
votes
1answer
68 views

How to simplify $\ln^2\left(x\right)+2 \ln x-3$

I dont know how to simplify $\ln^2\left(x\right)+2 \ln x-3$ I dont know how to get $(\ln(x)+1)(\ln(x)+3)$ But I am stuck and don't really know how to do that. I tried something like this: $2\ln ...
1
vote
5answers
92 views

What is the derivation for the derivative of $a^{t}$

Been driving me nuts. Can someone prove to me that $$\frac{d(a^t)}{dt} = a^t \ln(a)$$ Thank you!
5
votes
1answer
144 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
2
votes
4answers
298 views

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 $\ln x$.
0
votes
0answers
29 views

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 ...
1
vote
3answers
37 views

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.
2
votes
0answers
48 views

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 ...
0
votes
2answers
33 views

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 ...
1
vote
1answer
68 views

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 ...
4
votes
0answers
96 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
0
votes
2answers
80 views

Evaluate $\frac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$ with logarithmic differentiation

Spivak asks us to evaluate $$\dfrac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$$ by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to ...
1
vote
1answer
33 views

Differentiating logarithms

I am trying to prove that $$ f(x) = ^alog(x) => f'(x) = \frac {1} {ln(a)*x} $$ So I start at$$ f(x) = ^alog(x) $$ Then I move to:$$ f(x) =\frac {ln(x)} {ln(a)} $$ And there I get stuck: I want ...
2
votes
2answers
152 views

How to take the Limits of Logs

How would you take the limit of $$\frac{\log(n!)}{\log(n^n)}$$ as $n\rightarrow\infty$. I believe you have to remove the log raising it to their base. Is this correct ? Thanks.
1
vote
3answers
229 views

Derivative of $(\ln x)^{\ln x}$

How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, which functions? Thank you.
0
votes
2answers
22 views

Taking a logarithmic derivative of a function

I have the following expression: $ \log{\left(1 - \frac{r}{r_{s}} \right)} $ which I would like to take the following derivative of (and where $r_{s}$ is a constant): $ \frac{d\left(\log{\left(1 - ...
0
votes
1answer
21 views

Make derivations over sums

I have this kind of sums $$ \left(\sum_{i_1=0}^{4}\sum_{i_2=0}^{4}\log(f(X,i_1,i_2))\right)'\ $$ And we want to derive in respect to $${x_i}$$, which is an element of the vector X. How I should do ...
0
votes
3answers
79 views

logarithmic derivative of $x^{e^{(x^2+\cos x)}}$

I'm having a hard time taking the derivative of $$f(x) = x^{e^{(x^2+cosx)}}.$$ I'm aware that I have to take the logarithm of both sides. $$\ln(y) = \ln({x^{e^{(x^2+\cos x)}}}) = \ln(x)\cdot ...
1
vote
3answers
60 views

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 ...
0
votes
5answers
66 views

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 ...
2
votes
3answers
82 views

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 $\sec(x)$?
0
votes
0answers
30 views

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 ...
0
votes
2answers
61 views

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 $f(t) = \frac{87.5}{1 + 17.1e^{−0.91t}}$ where the ...
1
vote
1answer
71 views

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 ...
1
vote
1answer
40 views

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 ...
0
votes
2answers
127 views

Derivative of f(x)=arth(lnx)

I'm struggling with finding derivative of f(x)=arth(lnx). I've done following: x=th(y) ...
0
votes
1answer
36 views

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 ...
1
vote
0answers
73 views

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. ${d \operatorname{Tr}(\log(X))\over dX}=X^{-1}$, ...
2
votes
6answers
121 views

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.
2
votes
1answer
51 views

Derivative of $x\times n - 2^{\log_2 {x \times n}}$

I have a problem with solving derivative of $f(x)$ in this case: $$f(x) = x\times 10^9 - 2^{\log_{10} x\times 10^9}$$ This is what I have: $$f^\prime(x) = \lim_{m\to0} {f(x+m) - f(x)\over m}$$ $$= ...
0
votes
1answer
51 views

Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically

$\eqalign{ & y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr & \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr & {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 ...
4
votes
4answers
149 views

Differentiate $\log_{10}x$

My attempt: $\eqalign{ & \log_{10}x = {{\ln x} \over {\ln 10}} \cr & u = \ln x \cr & v = \ln 10 \cr & {{du} \over {dx}} = {1 \over x} \cr & {{dv} \over {dx}} ...
1
vote
4answers
65 views

Find the equation of the tangent to the curve $y = {2^x} + {2^{ - x}}$ at the point $(2,4{1 \over 4})$

$\eqalign{ & y = {2^x} + {2^{ - x}} \cr & \ln y = x\ln 2 - x\ln 2 \cr & \ln y = 0 \cr & {1 \over y}{{dy} \over {dx}} = 0 \cr & {{dy} \over {dx}} = 0 \cr} $ I've ...
1
vote
1answer
256 views

Comparing rates of change: which function increases faster?

I am comparing two functions for $x \ge 1$: $$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$ $$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...
1
vote
1answer
49 views

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 ...
4
votes
1answer
116 views

Need help understanding if a function is increasing or decreasing

I am working on understanding the following function: $$g(x) = \ln\Gamma\left(\frac{x}{4}\right) - \ln\Gamma\left(\frac{x}{5}+\frac{1}{2}\right) - \ln\Gamma\left(\frac{x}{20}+\frac{1}{2}\right) - ...
2
votes
1answer
88 views

The rate of increase of the Gamma Function over real numbers

If $$ x_1 > x_2 > 0$$ and $$\Delta{x}>0$$ does it follow that: $$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$ Would it be enough to show ...
2
votes
2answers
238 views

Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation

I am trying to get the lower bound for: $f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ...
3
votes
3answers
975 views

Finding the derivative of a function with a Natural Log.

I am trying to differentiate the function: $${\rm ln} \left(\frac{3x \ {\rm tan}(x)}{x^2 + 2}\right)$$ I think step one is to use the quotient rule of natural log expanding the expression. However ...
1
vote
1answer
102 views

Derivatives with Natural Log (Help)

This is the problem: $$f(x)=\ln[\sin(-2x)\cos(-2x)]$$ This is as far as I can get: $$\frac{-2[\cos(-2x)]}{\sin(-2x)}+\frac{2[\sin(-2x)]}{\cos(-2x)}$$ I'm familiar with the rules of differentiation ...
5
votes
2answers
168 views

Derivative of ${ x }^{ x }$ without logarithmic differentiation

With logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} ...
1
vote
2answers
86 views

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 ...
1
vote
4answers
201 views

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 ...