Find out the answer to "Find the indicated derivatives. dzdwifz=1w2−1" - Plainmath

glamrockqueen7

Answered question

2021-03-24

Find the indicated derivatives.

\frac{d z}{d w} if z = \frac{1}{\sqrt{w^{2} - 1}}

Answer & Explanation

Delorenzoz

Skilled2021-03-26Added 91 answers

Step 1: Given that:

z = \frac{1}{\sqrt{w^{2} - 1}}

We need to find the derivative of the given z-function
Step 2: Finding the Derivative

z = \frac{1}{\sqrt{w^{2} - 1}}

\Rightarrow z = {(w^{2} - 1)}^{- \frac{1}{2}}

Differentiate both sides w.r.t w we get,

\frac{d z}{d w} = \frac{d}{d w} ({(w^{2} - 1)}^{- \frac{1}{2}})

Using power rule,

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

, we have,

\frac{d z}{d w} = - \frac{1}{2} {(w^{2} - 1)}^{- \frac{1}{2} - 1} \times \frac{d}{d w} ((w^{2} - 1))

\Rightarrow \frac{d z}{d w} = - \frac{1}{2} {(w^{2} - 1)}^{- \frac{3}{2}} \times 2 w

\Rightarrow \frac{d z}{d w} = - w {(w^{2} - 1)}^{- \frac{3}{2}}

\Rightarrow \frac{d z}{d w} = - \frac{w}{{(w^{2} - 1)}^{\frac{3}{2}}}

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