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

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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Leonard Stokes
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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