# How do you compute the 200th derivative of f(x)=sin(2x)?

May 21, 2018

${2}^{200} \sin \left(2 x\right)$

#### Explanation:

$f \left(x\right) = \sin \left(2 x\right) \implies$
$\frac{d}{\mathrm{dx}} f \left(x\right) = 2 \cos \left(2 x\right) \implies$
${d}^{2} / {\mathrm{dx}}^{2} f \left(x\right) = - {2}^{2} \sin \left(2 x\right) \implies$
${d}^{3} / {\mathrm{dx}}^{3} f \left(x\right) = - {2}^{3} \cos \left(2 x\right) \implies$
${d}^{4} / {\mathrm{dx}}^{4} f \left(x\right) = {2}^{4} \sin \left(2 x\right) \implies$

This means that differentiating $f \left(x\right)$ 4 times results in the same function, with a multiplying factor of 4.

Differentiating 200 times is the same as repeating the above 50 times, and so

${d}^{200} / {\mathrm{dx}}^{200} f \left(x\right) = {\left({2}^{4}\right)}^{50} \sin \left(2 x\right) = {2}^{200} \sin \left(2 x\right)$