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What is the derivative of #log_10((x^2-1)/x)#?

Answer 1

Ezekiel Alexander

see below

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

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)#

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Answer 2

Christopher Allers

The derivative of ( \log_{10} \left( \frac{x^2 - 1}{x} \right) ) with respect to ( x ) is ( \frac{2x - 1}{x(x^2 - 1) \ln(10)} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.