Find dy/dt.
-$$y = ( 1 + \sin 9 t ) ^ { - 5 }$$
A) $- 45 ( \cos 9 t ) ^ { - 6 }$
B) $- 5 ( 1 + \sin 9 t ) ^ { - 6 } \cos 9 t$
C) $- 45 ( 1 + \sin 9 t ) ^ { - 6 } \cos 9 t$
D) $- 5 ( 1 + \sin 9 t ) ^ { - 6 }$

Question

Quizplus · Accepted Answer

Using chain rule, we have
$$\frac{dy}{dt} = \frac{dy}{du} \cdot \frac{du}{dt}$$
where $u = 1+\sin(9t)$.
Now,
$$\frac{dy}{du} = -5(1+\sin(9t))^{-6}$$
and
$$\frac{du}{dt} = 9\cos(9t)$$
Hence,
$$\frac{dy}{dt} = -5(1+\sin(9t))^{-6} \cdot 9\cos(9t)$$
Simplifying, we get
$$\frac{dy}{dt} = -45(1+\sin(9t))^{-6} \cos(9t)$$
Therefore, the answer is C.

Find Dy/dt y=(1+sin⁡9t)−5y = ( 1 + \sin 9 t ) ^ { - 5 }y=(1+sin9t)−5

Find Dy/dt $y = ( 1 + \sin 9 t ) ^ { - 5 }$