Higher-Order Derivatives of Trig Functions

The higher-order derivatives of [latex]\sin x[/latex] and [latex]\cos x[/latex] follow a repeating pattern. By following the pattern, we can find any higher-order derivative of [latex]\sin x[/latex] and [latex]\cos x[/latex].

Example: Finding Higher-Order Derivatives of [latex]y= \sin x[/latex]

Find the first four derivatives of [latex]y= \sin x[/latex].

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For [latex]y= \cos x[/latex], find [latex]\dfrac{d^4 y}{dx^4}[/latex].

Watch the following video to see the worked solution to Example: Finding Higher-Order Derivatives of [latex]y= \sin x[/latex] and the above Try It.

Example: Using the Pattern for Higher-Order Derivatives of [latex]y= \sin x[/latex]

Find [latex]\frac{d^{74}}{dx^{74}}(\sin x)[/latex].

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For [latex]y= \sin x[/latex], find [latex]\frac{d^{59}}{dx^{59}}(\sin x)[/latex].

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Example: An Application to Acceleration

A particle moves along a coordinate axis in such a way that its position at time [latex]t[/latex] is given by [latex]s(t)=2- \sin t[/latex].

Find [latex]v\left(\frac{\pi}{4}\right)[/latex]  and  [latex]a\left(\frac{\pi}{4}\right)[/latex]. Compare these values and decide whether the particle is speeding up or slowing down.

Watch the following video to see the worked solution to Example: An Application to Acceleration.

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A block attached to a spring is moving vertically. Its position at time [latex]t[/latex] is given by [latex]s(t)=2 \sin t[/latex].

Find [latex]v\left(\frac{5\pi}{6}\right)[/latex] and [latex]a\left(\frac{5\pi}{6}\right)[/latex]. Compare these values and decide whether the block is speeding up or slowing down.