Expert Answer to "Find the derivatives of the functionsz=4−3x3x2+x"

Deegan Chase

Answered question

2022-03-22

Find the derivatives of the functions

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

Answer & Explanation

Brennan Thompson

Beginner2022-03-23Added 10 answers

Step 1
The derivative of a function is the rate of change of the function with respect to the function variable. The derivative of a function of the form

\frac{f (x)}{g (x)}

can be calculated using the quotient rule, which is given by,

{(\frac{f (x)}{g (x)})}^{'} = \frac{g (x) \cdot f^{'} (x) - f (x) \cdot g^{'} (x)}{{(g (x))}^{2}}

Step 2
Therefore, the derivative of the function

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

is given by,

z^{'} = \frac{d (\frac{4 - 3 x}{3 x^{2} + x})}{d x}

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

= \frac{(3 x^{2} + x) \cdot (- 3) - (4 - 3 x) \cdot (6 x + 1)}{{(3 x^{2} + x)}^{2}}

= \frac{- 9 x^{2} - 3 x - (24 x + 4 - 18 x^{2} - 3 x)}{{(3 x^{2} + x)}^{2}}

= \frac{- 9 x^{2} - 3 x - 24 x - 4 + 18 x^{2} + 3 x}{{(3 x^{2} + x)}^{2}}

= \frac{9 x^{2} - 24 x - 4}{{(3 x^{2} + x)}^{2}}

Therefore, the derivative of the function is

\frac{9 x^{2} - 24 x - 4}{{(3 x^{2} + x)}^{2}}

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