Derivatives of Sinusoidal Functions

In today’s Graph Talks, the graph on the screen was $y=\sin x$ and the question was only this: where is it steepest? Your group pointed at the places the curve crosses its axis, and flat spots at the crests and troughs. Then you plotted those slopes as points in their own right — steepest, flat, steepest-downhill, flat — and a familiar shape assembled itself out of the slopes of sine. The derivative of $\sin x$ was hiding in plain sight:

$\text{If }f(x)=\sin x,\text{ then }{f}^{\prime }(x)=\cos x$

Where sine crests, cosine crosses zero — the flat top of the wave. Where sine crosses zero going up, cosine sits at its maximum — the steepest climb. The same slope-plotting move on $y=\cos x$ gives $-\sin x$: cosine starts at a crest, so its derivative starts flat and goes negative.¹

The four-step cycle

Differentiate repeatedly and the family chases its own tail:

$\sin x\to \cos x\to -\sin x\to -\cos x\to \sin x$

Four derivatives return you home. No other functions you have met do this, and it is the mathematical signature of things that oscillate — a pendulum’s displacement and velocity trade shapes exactly this way, which the motion-sensor demonstration made visible: the bob is fastest through the middle, momentarily still at the ends.

Combinations that model the world

Real oscillations arrive dressed: a tide is not $\sin t$ but something like $3\sin \left(\frac{\pi t}{6}\right)+5$. The chain rule from The Chain Rule handles the dressing — the inside $\frac{\pi t}{6}$ reports its rate $\frac{\pi }{6}$ as a factor — and the product rule handles genuine hybrids like $x\sin x$. That is all the machinery there is; the Smooth Landing task asks you to make a descent profile behave using exactly these moves, and Exponential and Sinusoidal Derivatives Practice works the same muscles on tides, daylight, and pendulums.

One habit to keep: before differentiating any sinusoidal model, predict where the rate should be zero — the crests and troughs of the story — and check your derivative against the prediction. A tide’s rate of rise at high tide had better come out to zero, and when it does, you know far more than “the algebra worked”.

Curriculum connection

A2.4

determine, through investigation using technology, the graph of the derivative $f^{\prime }(x)$ or $\frac{dy}{dx}$ of a given sinusoidal function [i.e., $f(x)=\sin x$, $f(x)=\cos x$] (e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of $x$ and graphing the ordered pairs; by using dynamic geometry software to verify graphically that when $f(x)=\sin x$, $f^{\prime }(x)=\cos x$, and when $f(x)=\cos x$, $f^{\prime }(x)=-\sin x$; by using a motion sensor to compare the displacement and velocity of a pendulum)

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A3.5

solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing $f(x)=\frac{x^2+1}{x-1}$ as the product $f(x)=(x^2+1)(x-1{)}^{-1}$], radical functions [e.g., by expressing $f(x)=\sqrt{x^2+5}$ as the power $f(x)=(x^2+5{)}^{1/2}$], and other simple combinations of functions [e.g., $f(x)=x\sin x$, $f(x)=\frac{\sin x}{\cos x}$]*

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B2.3

solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications (e.g., population growth, radioactive decay, temperature changes, hours of daylight, heights of tides), given the equation of a function* Sample problem: The size of a population of butterflies is given by the function $P(t)=\frac{6000}{1+49(0.6{)}^t}$ where $t$ is the time in days. Determine the rate of growth in the population after 5 days using the derivative, and verify graphically using technology.

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Footnotes

1. All of this is true in radians only — one more reason radians are the calculus-ready angle measure. Differentiate sine in degrees and an unlovely factor of $\frac{\pi }{180}$ leaks into every formula. ↩