Question

Find derivatives for the functions. Assume a, b, c, and k are constants. f(x)=\frac{a^{2}-x^{2}}{a^{2}+x^{2}}

Derivatives
ANSWERED
asked 2021-06-18
Find derivatives for the functions. Assume a, b, c, and k are constants.
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{a}^{{{2}}}-{x}^{{{2}}}}}{{{a}^{{{2}}}+{x}^{{{2}}}}}}\)

Expert Answers (1)

2021-06-19
First let's simplify f(x)
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{a}^{{{2}}}-{x}^{{{2}}}}}{{{a}^{{{2}}}+{x}^{{{2}}}}}}=-{\frac{{-{a}^{{{2}}}+{x}^{{{2}}}}}{{{a}^{{{2}}}+{x}^{{{2}}}}}}=-{\frac{{{a}^{{{2}}}+{x}^{{{2}}}-{2}{a}^{{{2}}}}}{{{a}^{{{2}}}+{x}^{{{2}}}}}}=-{1}+{2}{\frac{{{a}^{{{2}}}}}{{{a}^{{{2}}}+{x}^{{{2}}}}}}\)
Use the derivation rule
\(\displaystyle{\left({\frac{{{1}}}{{{g{{\left({x}\right)}}}'}}}=-{\frac{{{g}'{\left({x}\right)}}}{{{\left[{g{{\left({x}\right)}}}\right]}^{{{2}}}}}}\right.}\)
then
\(\displaystyle{f}'{\left({x}\right)}=-{2}{a}^{{{2}}}{\frac{{{\left({a}^{{{2}}}+{x}^{{{2}}}\right)}'}}{{{\left({a}^{{{2}}}+{x}^{{{2}}}\right)}^{{{2}}}}}}-{\frac{{{4}{a}^{{{a}}}{x}}}{{{\left({a}^{{{2}}}+{x}^{{{2}}}\right)}^{{{2}}}}}}\)
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