Recognize a repeating derivative pattern

Four presses return to the start

The curve starts as \(f(x) = \sin x\), and the faint curve is a copy of it. Each press of Differentiate once replaces the curve by its derivative, and the orange point marks its value at \(x = 0\).

After one press the curve is \(\cos x\), which is the old curve moved \(\frac{\pi}{2}\) to the left. After two presses it is \(-\sin x\), the start reflected in the x-axis, and after three presses it is \(-\cos x\).

After four presses the curve lies on the faint copy of \(\sin x\) again, and the values at \(x = 0\) are 0, 1, 0, \(-1\) and 0.

These four curves are the entries of a cycle that repeats every four presses. The order of a derivative is the number of presses, so only the remainder of the order divided by 4 matters.

A cycle and a constant factor

The curve now has a constant \(c\) inside: \(f(x) = \sin(cx)\). Each press still moves the curve one step around the cycle.

The dashed orange lines mark the height of the curve. Each press multiplies the height by \(c\), because the chain rule multiplies every derivative by \(c\), the derivative of the inside \(cx\).

After \(n\) presses the height is \(c^n\), and the remainder of \(n\) divided by 4 gives the entry of the cycle.

For \(e^{cx}\), the cycle has one entry, and each press multiplies the curve by \(c\).

Notation

1. Differentiate until the function returns, up to a constant factor.

   \[\begin{aligned} \sin x &\to \cos x \to -\sin x \\ &\to -\cos x \to \sin x \end{aligned}\]

2. The cycle has four entries. Divide the order \(n\) by 4, so that \(q\) full cycles leave a remainder \(r\), which picks the entry.

   \[n = 4q + r, \qquad \cy{f^{(n)}} = f^{(r)}\]

   A remainder of 0 means the original function.

3. By the chain rule, a constant \(c\) inside multiplies each derivative by \(c\), so \(n\) steps give the factor \(c^n\). The entry of the cycle keeps \(cx\) inside.

   \[\begin{aligned} \frac{d^n}{dx^n}\sin(cx) &= c^n \cdot \text{entry } r \\ \frac{d^n}{dx^n}e^{cx} &= c^n e^{cx} \end{aligned}\]

4. The pattern holds for \(\sin(cx)\), \(\cos(cx)\) and \(e^{cx}\), where \(c\) is a constant.

When it fails

The inside \(x^2\) is not a constant times \(x\). The first derivative is \(2x\cos(x^2)\), and the second is \(2\cos(x^2) - 4x^2\sin(x^2)\).

The derivative of the inside, \(2x\), changes with \(x\), so the factor is not the same at each step. After four presses the curve is far taller than the frame, so it does not return to a multiple of \(\sin(x^2)\). Differentiate one step at a time.

Example 1: a long order

Find the 50th derivative of \(f(x) = \cos x\).

1. Write the cycle for \(\cos x\). It has four entries, so the fourth derivative returns the curve to the start.

   \[\begin{aligned} \cos x &\to -\sin x \to -\cos x \\ &\to \sin x \to \cos x \end{aligned}\]

2. Divide 50 by 4 and keep the remainder. The first 48 derivatives are twelve full cycles, so they return the curve to the start.

   \[50 = 4 \cdot 12 + 2\]

3. The remainder is 2, so the answer is the entry two steps into the cycle.

   \[f^{(50)}(x) = -\cos x\]

Example 2: an exponential

Find the fifth derivative of \(f(x) = e^{-2x}\).

1. By the chain rule, each derivative multiplies by \(-2\), the derivative of the inside \(-2x\). After \(n\) derivatives the factor is \((-2)^n\).

   \[f^{(n)}(x) = (-2)^n e^{-2x}\]

2. A tempting wrong move: Ignore the sign and use \(2^5\).

   \[f^{(5)}(x) = 32\,e^{-2x}\]

   The factor \(-2\) changes the sign at each derivative, so after five derivatives the curve lies below the axis. The red curve lies above it.

3. Use the signed factor \((-2)^5 = -32\).

   \[f^{(5)}(x) = -32\,e^{-2x}\]

Practice

Find the sixth derivative of \(f(x) = \sin(3x)\).

The remainder of 6 divided by 4 picks the entry of the cycle. The label above the curve shows how each press multiplies the height by 3.

Questions

The figure shows \(\sin x\) after \(k\) presses of Differentiate once.

1. After four presses the curve lies on the start. What is \(f^{(8)}(x)\)?

   • \(\sin x\)
   • \(\cos x\)
   • \(-\sin x\)
   Answer

   \(\sin x\)

2. After three presses the value at \(x = 0\) is \(-1\). Which formula is the curve?

   • \(-\cos x\)
   • \(\cos x\)
   • \(-\sin x\)
   Answer

   \(-\cos x\)

3. Let \(f(x) = \cos(2x)\). What is \(f^{(4)}(0)\)?

   • 16
   • 1
   • 8
   • \(-16\)
   Answer

   16

Summary

You differentiate until the pattern repeats, divide the order by the length of the cycle, and keep the remainder.

This works because each full cycle returns the function, and a constant inside the function multiplies every step by the same factor.

You use it for high derivatives of sine, cosine and exponential functions, where the order is too large to differentiate step by step.

This is one of four skills in topic 3.6 of Which & Why: AP Calculus.