find the derivative of the function ( s=\frac{3}{7 pi} sin (7 t)+\frac{3}{4 pi} cos (4 t) ).\n\n( \frac{d…

find the derivative of the function ( s=\frac{3}{7 pi} sin (7 t)+\frac{3}{4 pi} cos (4 t) ).\n\n( \frac{d s}{d t}=square )\n(type an exact answer, using ( pi ) as needed.)

find the derivative of the function ( s=\frac{3}{7 pi} sin (7 t)+\frac{3}{4 pi} cos (4 t) ).\n\n( \frac{d s}{d t}=square )\n(type an exact answer, using ( pi ) as needed.)

Answer

Explanation:

Step1: Differentiate (\frac{3}{7\pi}\sin(7t))

Use the chain rule ((u(v(t)))^\prime = u^\prime(v(t))\cdot v^\prime(t)). Let (u = \sin(v)), (v = 7t). Then (u^\prime=\cos(v)), (v^\prime = 7). (\frac{d}{dt}\left(\frac{3}{7\pi}\sin(7t)\right)=\frac{3}{7\pi}\cdot\cos(7t)\cdot7=\frac{3}{\pi}\cos(7t))

Step2: Differentiate (\frac{3}{4\pi}\cos(4t))

Use the chain rule. Let (u=\cos(v)), (v = 4t). Then (u^\prime=-\sin(v)), (v^\prime = 4). (\frac{d}{dt}\left(\frac{3}{4\pi}\cos(4t)\right)=\frac{3}{4\pi}\cdot(-\sin(4t))\cdot4=-\frac{3}{\pi}\sin(4t))

Answer:

(\frac{3}{\pi}\cos(7t)-\frac{3}{\pi}\sin(4t))