Derivatives of Periodic Functions

  1. What is the 307th derivative of \(\sin(x)\)?

  2. Write down a formula for the derivative of each of the following functions. Note that you can check your derivative using any software capable of symbolic differentiation, e.g. WolframAlpha.

    \(\displaystyle a(x) = \sin(2x) \)
    \(\displaystyle b(x) = \cos(x-17) \)
    \(\displaystyle c(x) = \sin(x+2)+x+2 \)
    \(\displaystyle d(x) = 3x^2\sin(x) \)
    \(\displaystyle \varepsilon(x) = \tfrac{1}{x^2}\cos(x) \)
    \(\displaystyle f(x) = \cos(x)\sqrt{x} \)
    \(\displaystyle g(x) = \sin(x)\cos(x)\)
    \(\displaystyle h(x) = \cos\bigl(x^7\bigr) \)
    \(\displaystyle i(x) = \bigl(\cos(x)\bigr)^7 \)
    \(\displaystyle j(x) = x^4\sin(2x)\)
    \(\displaystyle k(x) = x^4+\cos(x)\)
    \(\displaystyle l(x) = \cos\bigl(\tfrac{2}{x}\bigr)\)
    \(\displaystyle m(x) = \cos\bigl(\ln(x)\bigr)\)
    \(\displaystyle n(x) = \ln\bigl(\cos(x)\bigr)\)
    \(\displaystyle o(x) = \ln(x)\cos(x)\)
    \(\displaystyle p(x) = \mathrm{e}^{\sin(x)}\)
    \(\displaystyle q(x) = \mathrm{e}^x\sin\bigl(x^\mathrm{e}\bigr)\)
    \(\displaystyle r(x) = \cos\bigl(\cos(x)\bigr)\)
    \(\displaystyle s(x) = \cos\Bigl(\cos\bigl(\cos(x)\bigr)\Bigr)\)
    \(\displaystyle t(x) = \sin(x)\sin\bigl(\cos(x)\bigr)\)
    \(\displaystyle u(x) = \mathrm{e}^{\cos(7x)}\)
    \(\displaystyle v(x) = \mathrm{e}\bigl(\cos(\mathrm{e}x)\bigr)^\mathrm{e}\)
    \(\displaystyle w(x) = \sqrt{\cos(3x)}\)
    \(\displaystyle \chi(x) = \frac{1}{\cos(x)}\)
    \(\displaystyle y(x) = \frac{\sin(x)}{\cos(x)}\)
    \(\displaystyle z(x) = \frac{\mathrm{e}^x}{\sin(3x)}\)