Expert Answer to "Find the first and second derivatives y=8x−7−6x"

e1s2kat26

Answered question

2021-10-21

Find the first and second derivatives

y = 8 x^{- 7} - \frac{6}{x}

Derrick

Skilled2021-10-22Added 94 answers

Step 1
Given:

y = 8 x^{- 7} - \frac{6}{x}

To Find: The first and second derivative of the given function.
Step 2
Solution:

y = 8 x^{- 7} - \frac{6}{x}

y = 8 x^{- 7} - 6 x^{- 1}

Differentiating with respect to x we get

\frac{d y}{d x} = \frac{d}{d x} (8 x^{- 7} - 6 x^{- 1})

\frac{d y}{d x} = 8 \frac{d}{d x} (x^{- 7}) - 6 \frac{d}{d x} (x^{- 1})

We know that

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

\Rightarrow \frac{d y}{d x} = 8 (- 7) x^{- 8} - 6 (- 1) (x^{- 2})

Hence

\frac{d y}{d x} = - 56 x^{- 8} + 6 x^{- 2}

Again differentiating with respect to x we get

\frac{d^{2} y}{{d x}^{2}} = \frac{d}{d x} (- 56 x^{- 8} + 6 x^{- 2})

= - 56 \frac{d}{d x} (x^{- 8}) + 6 \frac{d}{d x} (x^{- 2})

= - 56 (- 8) x^{- 9} + 6 (- 2) x^{- 3}

= 448 x^{- 9} - 12 x^{- 3}

Hence

\frac{d^{2} y}{{d x}^{2}} = 448 x^{- 9} - 12 x^{- 3}

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