The higher-order derivatives of sinx and cosx follow a repeating pattern. By following the pattern, we can find any higher-order derivative of sinx and cosx.
Example: Finding Higher-Order Derivatives of y=sinx
Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of sinx equals sinx, so
Watch the following video to see the worked solution to Example: Finding Higher-Order Derivatives of y=sinx and the above Try It.
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A particle moves along a coordinate axis in such a way that its position at time t is given by s(t)=2−sint.
Find v(π4) and a(π4). Compare these values and decide whether the particle is speeding up or slowing down.
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First find v(t)=s′(t): v(t)=s′(t)=−cost. Thus, v(π4)=−1√2.
Next, find a(t)=v′(t).
Thus, a(t)=v′(t)=sint and we have a(π4)=1√2.
Since v(π4)=−1√2<0 and a(π4)=1√2>0, we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is traveling.
Consequently, the particle is slowing down.
Watch the following video to see the worked solution to Example: An Application to Acceleration.
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