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How do you find the derivative of rational functions: #(3/x)#, #(7/x^2)#, #(x^3/sqrt(x))#?

Answer 1

Evelyn Agard

You can use the Quotient Rule or rewrite them as powers, as in:

#3/x=3x^-1# Derived: #3*(-1)*x^(-1-1)=-3x^-2=-3/x^2#

and:

#7/x^2=7x^-2# Derived: #7*(-2)*x^(-2-1)=-14x^-3=-14/x^3#

and:

#x^3/sqrt(x)=x^3*x^(-1/2)=x^(5/2)# Derived: #5/2x^(5/2-1)=5/2x^(3/2)=5/2xsqrt(x)#

hope it helps

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Answer 2

Emma Aldridge

To find the derivative of rational functions:

( \frac{d}{dx} \left(\frac{3}{x}\right) )
The derivative is ( -\frac{3}{x^2} ).
( \frac{d}{dx} \left(\frac{7}{x^2}\right) )
The derivative is ( \frac{14}{x^3} ).
( \frac{d}{dx} \left(\frac{x^3}{\sqrt{x}}\right) )
The derivative is ( \frac{3x^2}{2\sqrt{x}} - \frac{x^3}{2x^{3/2}} ).

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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.