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How do you find the derivative of #f(x) = log_x (3)#?

Answer 1

Emily Ammerman

#(df)/(dx)(x) = -log_e3/(x log_e^2(x))#

#y = log_x 3 = (log_b 3)/(log_b x)# for any convenient basis #b#

Calling now #g(x,y) = y log_e x - log_e 3 = 0# after taking #b = e#, we have

#dg = g_x dx + g_y dy = 0#

then

#(dy)/(dx) = - (g_x)/(g_y) = -((y/x))/(log_e x) = -y/(x log_e(x)) = -log_e3/(x log_e^2(x))#

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Answer 2

Christopher Allers

To find the derivative of ( f(x) = \log_x(3) ), you can use the logarithmic differentiation formula:

[ f'(x) = \frac{d}{dx}\left(\log_x(3)\right) = \frac{1}{\ln(x)} \cdot \frac{d}{dx}(3) ]

Since ( \log_x(3) ) is a constant with respect to ( x ), its derivative is zero:

[ \frac{d}{dx}(3) = 0 ]

Therefore, the derivative of ( f(x) = \log_x(3) ) is:

[ f'(x) = 0 ]

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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.