Question: "Find the first and second derivatives y=4−2x−x−3" - Plainmath

Globokim8

Answered question

2021-10-14

Find the first and second derivatives

y = 4 - 2 x - x^{- 3}

Answer & Explanation

Tasneem Almond

Skilled2021-10-15Added 91 answers

Step 1
Given:

y = 4 - 2 x - x^{- 3}

Using:
Sum/Difference rule:

{(f \pm g)}^{'} = f^{'} \pm g^{'}

Power rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

Step 2
Now,
First derivative is given by:

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

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

= 0 - 2 - (- 3 x^{- 3 - 1})

= - 2 + 3 x^{- 4}

And
The second derivative is given by:

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

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

= 0 + 3 \cdot (- 4) x^{- 4 - 1}

= - 12 x^{- 5}

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