"Find derivatives for the functions. Assume a, b,..." solved and explained - Plainmath

ankarskogC

Answered question

2021-06-18

Find derivatives for the functions. Assume a, b, c, and k are constants.

f (x) = \frac{a^{2} - x^{2}}{a^{2} + x^{2}}

Answer & Explanation

SoosteethicU

Skilled2021-06-19Added 102 answers

First let's simplify f(x)

f (x) = \frac{a^{2} - x^{2}}{a^{2} + x^{2}} = - \frac{- a^{2} + x^{2}}{a^{2} + x^{2}} = - \frac{a^{2} + x^{2} - 2 a^{2}}{a^{2} + x^{2}} = - 1 + 2 \frac{a^{2}}{a^{2} + x^{2}}

Use the derivation rule

(\frac{1}{{g (x)}^{'}} = - \frac{g^{'} (x)}{{[g (x)]}^{2}}

then

f^{'} (x) = - 2 a^{2} \frac{{(a^{2} + x^{2})}^{'}}{{(a^{2} + x^{2})}^{2}} - \frac{4 a^{a} x}{{(a^{2} + x^{2})}^{2}}

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