Expert Answer to "Find the derivatives without using product or quotient..."

Ann Tice

Answered question

2021-12-03

Find the derivatives without using product or quotient rules.

y = \frac{x^{3} + 8}{x^{2} - 2 x + 4}

Clara Clark

Beginner2021-12-04Added 17 answers

Step 1
Consider the given equation

y = \frac{x^{3} + 8}{x^{2} - 2 x + 4}

Step 2
First factorise the numerator of the given function as shown below.

x^{3} + 8 = (x + 2) (x^{2} - 2 x + 4)

Step 3
The derivative of the given function is calculated as follows.

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

= \frac{d}{d x} (x + 2)

= \frac{d}{d x} (x) + \frac{d}{d x} (2)

=1+0
=1
Step 4
Therefore, the derivative of the given function

\frac{d}{d x} (\frac{x^{3} + 8}{x^{2} - 2 x + 4}) = 1

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