What is the derivative of ${log}_{3} (\frac{x \sqrt{x - 1}}{2})$ $log_3((xsqrt(x-1))/2)$ ?

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Answer 1 · 2017-03-31T20:39:43

see below

Use the following Properties of Logarithm to expand the problem before taking derivatives.

${log}_{b} (x y) = {log}_{b} x + {log}_{b} y$ $color(red)(log_b(xy)=log_bx+log_by$
${log}_{b} (\frac{x}{y}) = {log}_{b} x - {log}_{b} y$ $color(red)(log_b(x/y)=log_bx-log_by$
${log}_{b} x^{n} = n {log}_{b} x$ $color(red)(log_b x^n =n log_bx$

Use the formula $\frac{d}{d x} ({log}_{b} x) = \frac{1}{x ln b}$ $color(red)(d/dx(log_bx)=1/(xln b)$ to find the derivative

$y = {log}_{3} (\frac{x \sqrt{x - 1}}{2}) = {log}_{3} (\frac{x {(x - 1)}^{\frac{1}{2}}}{2})$ $y=log_3((xsqrt(x-1))/2)=log_3((x(x-1)^(1/2))/2)$

$y = {log}_{3} x + {log}_{3} {(x - 1)}^{\frac{1}{2}} - {log}_{3} 2$ $y=log_3x+log_3(x-1)^(1/2)-log_3 2$

$= {log}_{3} x + \frac{1}{2} {log}_{3} (x - 1) - {log}_{3} 2$ $=log_3x+1/2 log_3(x-1)-log_3 2$

$color(blue)(y'=1/(xln3)+1/2 *1/(x-1)-0$

$color(blue)(y'=1/(xln3)+1/(2x-2)$

What is the derivative of log_3((xsqrt(x-1))/2)log3(x√x−12)log_3((xsqrt(x-1))/2)?