Calculate the first five derivatives of f(x) = \sin x. Then determine f^{(9)}(x)\ and\ f^{(102)}(x).

Emeli Hagan 2021-06-07 Answered
Calculate the first five derivatives of \(\displaystyle{f{{\left({x}\right)}}}={\sin{{x}}}\). Then determine \(\displaystyle{{f}^{{{\left({9}\right)}}}{\left({x}\right)}}\ {\quad\text{and}\quad}\ {{f}^{{{\left({102}\right)}}}{\left({x}\right)}}\).

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Expert Answer

Leonard Stokes
Answered 2021-06-08 Author has 16727 answers
Step 1
Given
\(\displaystyle{f{{\left({x}\right)}}}={\sin{{x}}}\)
The first five derivatives are
\(\displaystyle{f}'{\left({x}\right)}={\cos{{x}}}\)
\(\displaystyle{f}{''}{\left({x}\right)}=-{\sin{{x}}}\)
\(\displaystyle{f}^{{{\left({3}\right)}}}=-{\cos{{x}}}\)
\(\displaystyle{f}^{{{\left({4}\right)}}}={\sin{{x}}}\)
\(\displaystyle{f}^{{{\left({5}\right)}}}={\cos{{x}}}\)
From this it can be concluded that every fourth one repeats .
Step 2
The eighth derivative will be
\(\displaystyle{{f}^{{{\left({8}\right)}}}{\left({x}\right)}}={\sin{{x}}}\)
so the ninth derivative will be
\(\displaystyle{{f}^{{{\left({9}\right)}}}{\left({x}\right)}}={\cos{{x}}}\)
The hundredth derivative will be
\(\displaystyle{{f}^{{{\left({100}\right)}}}{\left({x}\right)}}={\sin{{x}}}\)
Then,
\(\displaystyle{{f}^{{{\left({102}\right)}}}{\left({x}\right)}}=-{\sin{{x}}}\)
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Answered 2021-12-25 Author has 11052 answers

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