3.5 Derivatives of Trigonometric Functions

Derivatives of Trigonometric Functions

Derivative rules for sine and cosine

The derivatives of sine and cosine form a tight pair: each one's derivative leads to the other (with a sign change for cosine). These two rules are the foundation for differentiating all six trig functions.

• $\frac{d}{dx} \sin(x) = \cos(x)$
  • At $x = 0$, the sine curve has slope $\cos(0) = 1$. At $x = \frac{\pi}{2}$, the slope is $\cos\!\left(\frac{\pi}{2}\right) = 0$, which matches the peak of the sine wave.
• $\frac{d}{dx} \cos(x) = -\sin(x)$
  • The negative sign matters. At $x = 0$, the cosine curve is at its maximum, so its slope is $-\sin(0) = 0$, exactly what you'd expect at a peak.

With the chain rule: When the argument is something other than plain $x$, you multiply by the derivative of the inner function.

• If the argument is a constant multiple $ax$:
  • $\frac{d}{dx} \sin(ax) = a\cos(ax)$
  • $\frac{d}{dx} \cos(ax) = -a\sin(ax)$
• If the argument is a general function $u(x)$:
  • $\frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx}$
  • $\frac{d}{dx} \cos(u) = -\sin(u) \cdot \frac{du}{dx}$

For example, to differentiate $\sin(x^2)$, treat $u = x^2$ so $\frac{du}{dx} = 2x$. The result is $2x\cos(x^2)$.

Derivatives of the other four trig functions

Each of these can be derived from sine and cosine using the quotient rule, but you should memorize the results. Notice the pattern: the "co-" functions (cosine, cotangent, cosecant) all pick up a negative sign.

• $\frac{d}{dx} \tan(x) = \sec^2(x)$
• $\frac{d}{dx} \cot(x) = -\csc^2(x)$
• $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$
• $\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)$

With the chain rule: The same logic applies. Multiply by $\frac{du}{dx}$.

• $\frac{d}{dx} \tan(u) = \sec^2(u) \cdot \frac{du}{dx}$
• $\frac{d}{dx} \cot(u) = -\csc^2(u) \cdot \frac{du}{dx}$
• $\frac{d}{dx} \sec(u) = \sec(u)\tan(u) \cdot \frac{du}{dx}$
• $\frac{d}{dx} \csc(u) = -\csc(u)\cot(u) \cdot \frac{du}{dx}$

For example, $\frac{d}{dx} \sec(3x) = 3\sec(3x)\tan(3x)$.

Higher-order derivatives of sine and cosine

Taking repeated derivatives of sine and cosine produces a four-step cycle. This is worth memorizing because it shows up in differential equations and Taylor series later on.

For $\sin(x)$:

1. First derivative: $\cos(x)$
2. Second derivative: $-\sin(x)$
3. Third derivative: $-\cos(x)$
4. Fourth derivative: $\sin(x)$ (back to the start)

For $\cos(x)$:

1. First derivative: $-\sin(x)$
2. Second derivative: $-\cos(x)$
3. Third derivative: $\sin(x)$
4. Fourth derivative: $\cos(x)$ (back to the start)

To find the $n$th derivative without computing every step, divide $n$ by 4 and use the remainder. For $\sin(x)$: remainder 0 gives $\sin(x)$, remainder 1 gives $\cos(x)$, remainder 2 gives $-\sin(x)$, remainder 3 gives $-\cos(x)$.

The second derivative is also useful for concavity. Since $\frac{d^2}{dx^2}\sin(x) = -\sin(x)$, the sine curve is concave down wherever $\sin(x) > 0$ and concave up wherever $\sin(x) < 0$, with inflection points at every integer multiple of $\pi$.

Characteristics of Trigonometric Functions

These terms come up constantly when you're working with trig derivatives, so make sure they're solid:

• Period: The length of one full cycle. For $\sin(x)$ and $\cos(x)$, the period is $2\pi$. For $\sin(bx)$, the period is $\frac{2\pi}{b}$.
• Amplitude: The maximum displacement from the midline. For $A\sin(x)$, the amplitude is $|A|$.
• Frequency: The number of cycles per unit interval, which is the reciprocal of the period.
• Radian: The standard angle unit in calculus. All the derivative formulas above assume the input is in radians. If you use degrees, the formulas break.