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How do you find the derivative of #y = (2x+3)^4 / x#?

Answer 1

Aurora Alvarado

#[(2x+3)^3 (8x^2 -2x-3)]/x^2#

By applying the quotient rule, the derivative would be

#[4(2x+3)^3 (2x)(x) - (2x+3)^4]/x^2#

=#[(2x+3)^3 (8x^2 -2x-3)]/x^2#

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Answer 2

Cameron Alexander

#([8x * (2x+3)^3] - [(2x+3)^4]) / x^2#

Using quotient rule,

#d/dx (2x+3)^4/x = ([8x * (2x+3)^3] - [(2x+3)^4]) / x^2#

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Answer 3

Emily Ammerman

To find the derivative of ( y = \frac{(2x + 3)^4}{x} ), you can use the quotient rule. Here are the steps:

Identify the numerator and denominator functions. Let ( u = (2x + 3)^4 ) and ( v = x ).
Find the derivatives ( u' ) and ( v' ). ( u' = 4(2x + 3)^3 \cdot 2 ) and ( v' = 1 ).
Apply the quotient rule formula: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} ).
Substitute the values: ( \frac{(4(2x + 3)^3 \cdot 2 \cdot x) - ((2x + 3)^4 \cdot 1)}{x^2} ).
Simplify: ( \frac{8x(2x + 3)^3 - (2x + 3)^4}{x^2} ).

So, the derivative of ( y ) with respect to ( x ) is ( \frac{8x(2x + 3)^3 - (2x + 3)^4}{x^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.