Expert answers to "Find the first and second derivatives w=3z7−7z3+21z2"

protisvitfc

Answered question

2021-11-17

Find the first and second derivatives

w = 3 z^{7} - 7 z^{3} + 21 z^{2}

Robert Harris

Beginner2021-11-18Added 23 answers

Step 1
Given:

w = 3 z^{7} - 7 z^{3} + 21 z^{2}

To find first and second derivative of the function.
Solution:
Power rule of differentiation states that:

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

Differentiating the given function with respect to x,

\frac{d w}{d z} = \frac{d}{d z} (3 z^{7} - 7 z^{3} + 21 z^{2})

= 3 \cdot 7 z^{7 - 1} - 7 \cdot 3 z^{3 - 1} + 21 \cdot 2 z^{2 - 1}

= 21 z^{6} - 21 z^{2} + 42 z

Step 2
Again differentiating with respect to z,

\frac{d^{2} w}{{d z}^{2}} = \frac{d}{d z} (\frac{d w}{d z})

= \frac{d}{d z} (21 z^{6} - 21 z^{2} + 42 z)

= 21 \cdot 6 z^{6 - 1} - 21 \cdot 2 z^{2 - 1} + 42 \cdot 1

= 126 z^{5} - 42 z + 42

Hence,

\frac{d^{2} w}{{d z}^{2}} = 126 z^{5} - 42 z + 42

Alrew1959

Beginner2021-11-19Added 16 answers

w = 3 z^{7} - 7 z^{3} + 21 z^{2}

Apply the Sum/Difference Rule:

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

= \frac{d}{d z} (3 z^{7}) - \frac{d}{d z} (7 z^{3}) + \frac{d}{d z} (21 z^{2})

\frac{d}{d z} (3 z^{7}) = 21 z^{6}

\frac{d}{d z} (6 z^{3}) = 21 z^{2}

\frac{d}{d z} (21 z^{2}) = 42 z

= 21 z^{6} - 21 z^{2} + 42 z

\frac{d}{d z} (21 z^{6} - 21 z^{2} + 42 z)

Apply the Sum/Difference Rule:

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

= \frac{d}{d z} (21 z^{6}) - \frac{d}{d z} (21 z^{2}) + \frac{d}{d z} (42 z)

\frac{d}{d z} (21 z^{6}) = 126 z^{5}

\frac{d}{d z} (21 z^{2}) = 42 z

\frac{d}{d z} (42 z) = 42

= 126^{5} - 42 z + 42

Jeffrey Jordon

Expert2022-03-13Added 2605 answers

Answer is given below (on video)

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