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How do you use the quotient rule to find the derivative of #y=(ax+b)/(cx+d)# ?

Answer 1

Ezekiel Alexander

#y'=(ad-bc)/(cx+d)^2#

Quotient Rule is used to

#y=f(x)/g(x)#, then #y'=(f'(x)g(x)−f(x)g'(x))/(g(x))^2#

#y=(ax+b)/(cx+d)#

separating the two sides in relation to x,

#y'=((ax+b)'(cx+d)-(ax+b)(cx+d)')/(cx+d)^2#

#y'=(a(cx+d)-(ax+b)c)/(cx+d)^2#

#y'=(ad-bc)/(cx+d)^2#

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Answer 2

Axel Alvarez

To find the derivative of ( y = \frac{ax + b}{cx + d} ) using the quotient rule:

Identify the functions ( u ) and ( v ) as ( u(x) = ax + b ) and ( v(x) = cx + d ).
Apply the quotient rule, which states that if ( y = \frac{u}{v} ), then ( y' = \frac{u'v - uv'}{v^2} ).
Find the derivatives ( u' ) and ( v' ) using the power rule for differentiation.
Substitute the values into the quotient rule formula.
Simplify the expression to get the derivative of ( y ).

The derivative of ( y ) with respect to ( x ) is: [ y' = \frac{(a)(cx + d) - (ax + b)(c)}{(cx + d)^2} ]

Simplify the numerator and denominator as needed.

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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.