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How do you find the derivative of # sqrt(x)/(x^3+1)#?

Answer 1

Luke Alvarez

#(df)/(dx)=(1-5x^3)/(2sqrtx(x^3+1)^2)#

As #f(x)=sqrtx/(x^3+1)#, using quotient rule,

#(df)/(dx)=(d/(dx)sqrtx xx (x^3+1)-d/(dx)(x^3+1)xx sqrtx)/(x^3+1)^2#

= #(1/(2sqrtx)xx(x^3+1)-3x^2 xx sqrtx)/(x^3+1)^2#

= #(x^3+1-6x^3)/(2sqrtx(x^3+1)^2)#

= #(1-5x^3)/(2sqrtx(x^3+1)^2)#

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Answer 2

Christopher Allers

To find the derivative of ( \frac{\sqrt{x}}{x^3+1} ), you can use the quotient rule, which states that if you have a function ( \frac{f(x)}{g(x)} ), then the derivative is given by ( \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} ). Apply the rule to the given function.

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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.