Higher-Order Derivatives of Trig Functions
The higher-order derivatives of [latex]\sin x[/latex] and [latex]\cos x[/latex] exhibit a cyclical pattern, making it possible to predict any higher-order derivative of these functions. By understanding this repeating sequence, you can easily compute derivatives beyond the first order.
To illustrate, let’s calculate the first four derivatives of [latex]y= \sin x[/latex].
[latex]\begin{array}{ll}
\text{Start with the function itself:} & y = \sin x \\
\text{The first derivative of } \sin x \text{ is:} & \dfrac{dy}{dx} = \cos x \\
\text{The second derivative becomes:} & \dfrac{d^2y}{dx^2} = -\sin x \\
\text{Continuing, the third derivative is:} & \dfrac{d^3y}{dx^3} = -\cos x \\
\text{The fourth derivative brings us back to the starting function:} & \dfrac{d^4y}{dx^4} = \sin x
\end{array}[/latex]
This sequence of derivatives demonstrates a pattern that repeats every four derivatives.
- [latex]\sin x[/latex] leads to [latex]\cos x[/latex]
- [latex]\cos x[/latex] leads to [latex]- \sin x[/latex]
- [latex]- \sin x[/latex] leads to [latex]- \cos x[/latex]
- [latex]- \cos x[/latex] leads back to [latex]\sin x[/latex]
Understanding this cyclical pattern not only simplifies calculations but also equips us with a systematic approach for determining any higher-order derivative.
How to: Use the Cyclic Pattern to Determine Higher-Order Derivatives of Sine and Cosine Functions
- Determine the order of the derivative you need (let’s call it [latex]n[/latex]).
- Calculate the remainder when [latex]n[/latex] is divided by [latex]4[/latex]. The remainder determines the position in the cycle:
- Remainder 0: The derivative returns to the original function.
- Remainder 1: The derivative progresses to the next function in the cycle.
- Remainder 2: The derivative is the negative of the original function.
- Remainder 3: The derivative is the negative of the next function in the cycle.
- For [latex]\sin x[/latex]:
- If [latex]n[/latex] is divisible by [latex]4[/latex] (remainder [latex]0[/latex]), the [latex]n[/latex]-th derivative is [latex]\sin x[/latex].
- If [latex]n[/latex] divided by [latex]4[/latex] gives a remainder of [latex]1[/latex], it is [latex]\cos x[/latex].
- If the remainder is [latex]2[/latex], it is [latex]- \sin x[/latex].
- If the remainder is [latex]3[/latex], it is [latex]- \cos x[/latex].
- For [latex]\cos x[/latex]:
- Remainder [latex]0[/latex]: [latex]\cos x[/latex]
- Remainder [latex]1[/latex]: [latex]- \sin x[/latex]
- Remainder [latex]2[/latex]: [latex]- \cos x[/latex]
- Remainder [latex]3[/latex]: [latex]\sin x[/latex]
Find [latex]\dfrac{d^{74}}{dx^{74}}(\sin x)[/latex].
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(\dfrac{\pi}{4}\right)[/latex] and [latex]a\left(\dfrac{\pi}{4}\right)[/latex]. Compare these values and decide whether the particle is speeding up or slowing down.