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How do you find the derivative of #4x/(4+x^2)#?

Answer 1

Luke Alvarez

Use quotient rule, the derivative would be:

#4{1.(4+x^2)- x(2x)}/(4+x^2)^2#

= #4(4-x^2)/(4+x^2)^2#

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Answer 2

Scarlett Allison

To find the derivative of ( \frac{4x}{4+x^2} ), you can use the quotient rule, which states that the derivative of ( \frac{u}{v} ) is given by ( \frac{vu' - uv'}{v^2} ), where ( u ) and ( v ) are functions of ( x ). Applying the quotient rule to the given function:

( u = 4x )
( v = 4 + x^2 )

( u' = 4 )
( v' = 2x )

Now, using the quotient rule:

( \frac{d}{dx} \left( \frac{4x}{4+x^2} \right) = \frac{(4+x^2)(4) - (4x)(2x)}{(4+x^2)^2} )

( = \frac{(4 + 4x^2) - (8x^2)}{(4+x^2)^2} )

( = \frac{4 + 4x^2 - 8x^2}{(4+x^2)^2} )

( = \frac{4 - 4x^2}{(4+x^2)^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.