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What is the derivative of #sqrt(5x+sqrt(5x+sqrt(5x)))# ?

Answer 1

Ezekiel Alexander

#d/(dx) sqrt(5x+sqrt(5x+sqrt(5x)))#

#= 1/(2sqrt(5x+sqrt(5x+sqrt(5x)))) (5+1/(2sqrt(5x+sqrt(5x))) (5+ 5/(2 sqrt(5x))))#

#d/(dx) sqrt(5x+sqrt(5x+sqrt(5x)))#

#= d/(dx) (5x+(5x+(5x)^(1/2))^(1/2))^(1/2)#

#= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * d/(dx) (5x+(5x+(5x)^(1/2))^(1/2))#

#= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * (5+1/2 (5x+(5x)^(1/2))^(-1/2) * d/(dx) (5x+(5x)^(1/2)))#

#= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * (5+1/2 (5x+(5x)^(1/2))^(-1/2) * (5+ 1/2 (5x)^(-1/2) * d/(dx) (5x)))#

#= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * (5+1/2 (5x+(5x)^(1/2))^(-1/2) * (5+ 1/2 (5x)^(-1/2) * 5))#

#= 1/(2sqrt(5x+sqrt(5x+sqrt(5x)))) (5+1/(2sqrt(5x+sqrt(5x))) (5+ 5/(2 sqrt(5x))))#

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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.