What is the derivative of ${log}_{10} (\frac{x^{2} - 1}{x})$ $log_10((x^2-1)/x)$ ?

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Answer 1 · 2017-03-31T20:44:57

see below

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

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

Then use the formula $color(red)(d/dx(log_bf(x))=1/(f(x)ln b) * f'(x)$ to find the derivative

$y=log_10((x^2-1)/x)$

$=log_10(x^2-1)-log_10 x$

$color(blue)(y'=1/((x^2-1)ln10)*2x-1/(xln10)$

$color(blue)(y'=(2x)/((x^2-1)ln10)-1/(xln10)$

What is the derivative of log_10((x^2-1)/x)log10(x2−1x)log_10((x^2-1)/x)?