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What is the derivative of #f(x)=sqrt(1+log_3(x)# ?

Answer 1

William Abdalla

#d/dx(sqrt(1+log_3x))#

#=((d/dx)(1+log_3x))/{2sqrt(1+log_3x)}#

#=((d/dx)(1+logx/log3))/{2sqrt(1+log_3x)}#

#=(1/(xln3))/{2sqrt(1+log_3x)}#

#=1/(2xln3sqrt(1+log_3))#

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Answer 2

Axel Alvarez

To find the derivative of ( f(x) = \sqrt{1 + \log_3(x)} ), we can use the chain rule.

[ f'(x) = \frac{1}{2\sqrt{1 + \log_3(x)}} \cdot \frac{d}{dx}(1 + \log_3(x)) ]

[ = \frac{1}{2\sqrt{1 + \log_3(x)}} \cdot \frac{d}{dx}(\log_3(x) + 1) ]

[ = \frac{1}{2\sqrt{1 + \log_3(x)}} \cdot \frac{1}{x\ln(3)} ]

[ = \frac{1}{2x\sqrt{1 + \log_3(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.