How do you find the derivative of #f(x) = log_x 2#?

Answer 1

Evelyn Agard

Change the base: #log_x 2 = (ln2)/(lnx)#

#f(x) = log_x 2 = (ln2)/(lnx)#

I think maybe the quotient rule is clearest rather than rewriting any more.

#f'(x) = ((0)(lnx)-(ln2)(1/x))/(lnx)^2#

# = -ln2/(x(lnx)^2)#

If wished, we can rewrite again:

# = -(ln2)/lnx 1/(xlnx)#

# = -log_x 2/(xlnx)#

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Answer 2

Aurora Alvarado

To find the derivative of ( f(x) = \log_x 2 ), you can use the following steps:

Apply the change of base formula for logarithms: ( \log_x 2 = \frac{\ln 2}{\ln x} ).
Differentiate the expression ( \frac{\ln 2}{\ln x} ) with respect to ( x ) using the quotient rule.

Following these steps, you'll find the derivative of ( f(x) = \log_x 2 ) to be:

[ f'(x) = -\frac{\ln 2}{(\ln x)^2} \cdot \frac{1}{x} ]

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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.