Derivatives of a fraction functionAn example of a fraction function is: y = − 8...

uri2e4g

uri2e4g

Answered

2022-07-09

Derivatives of a fraction function
An example of a fraction function is:
y = 8 x ( x 2 + 3 ) 2
The quotient rule says that if the function one wishes to differentiate, f ( x ), can be written as:
h ( x ) = f ( x ) g ( x )
Then the derivative is (according to what I learned):
h ( x ) = f ( x ) g ( x ) f ( x ) g ( x ) ( g ( x ) ) 2
Then I think the procedure is the following:
y = 24 ( x 2 1 ) ( ( x 2 + 3 ) 2 ) 2 = 24 ( x 2 1 ) ( x 2 + 3 ) 4
However, the solution is...
y = 24 ( x 2 1 ) ( x 2 + 3 ) 3
What are my mistakes?
What is the correct way to derivate fractions?

Answer & Explanation

Jayvion Tyler

Jayvion Tyler

Expert

2022-07-10Added 23 answers

Your second step (after writing down the quotient rule) should be:
y = 8 ( x 2 + 3 ) 2 + 8 x 2 2 x ( x 2 + 3 ) ( ( x 2 + 3 ) 2 ) 2 ,
and then an x 2 + 3 cancels off and gives you the correct answer:
8 ( x 2 + 3 ) 2 + 8 x 2 2 x ( x 2 + 3 ) ( ( x 2 + 3 ) 2 ) 2 = 8 ( x 2 + 3 ) + 32 x 2 ( x 2 + 3 ) 3 = 24 ( x 2 1 ) ( x 2 + 3 ) 3 .
Rebecca Villa

Rebecca Villa

Expert

2022-07-11Added 3 answers

Hint #1: If you write your fraction as
H I G H / L O W ,
then the derivative for the quotient is "given" by the mnemonic
L O W d ( H I G H ) H I G H d ( L O W ) ( L O W ) 2 .
Hint #2: d ( H I G H ) = 8
Hint #3: d ( L O W ) = 2 ( x 2 + 3 ) ( 2 x ) = 4 x ( x 2 + 3 )
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