DIFFERENTIATION 10

Trigonometric Derivatives

Sine, cosine and tangent have simple derivative rules that connect directly to their geometric behaviour.

The main idea

The three basic derivatives are

\[ \boxed{ \frac{d}{dx}\sin x=\cos x } \] \[ \boxed{ \frac{d}{dx}\cos x=-\sin x } \] \[ \boxed{ \frac{d}{dx}\tan x=\sec^2 x } \]

These formulas assume that angles are measured in radians.

1. Derivative of sine

\[ y=\sin x, \]

then

\[ \frac{dy}{dx}=\cos x. \]

So the rate of change of sine is given by cosine.

For example, at

\[ x=0, \]

we have

\[ \frac{dy}{dx} = \cos 0 = 1. \]

So the sine curve has gradient \(1\) at \(x=0\).

2. Derivative of cosine

\[ y=\cos x, \]

then

\[ \frac{dy}{dx}=-\sin x. \]

\[ x=0, \]

we have

\[ \frac{dy}{dx}=0. \]

So the cosine curve has a horizontal tangent at \(x=0\).

3. Derivative of tangent

Since

\[ \tan x=\frac{\sin x}{\cos x}, \]

we can differentiate using the quotient rule.

This gives

\[ \frac{d}{dx}\tan x = \frac{ \cos^2 x+\sin^2 x }{ \cos^2 x }. \]

Using

\[ \sin^2x+\cos^2x=1, \]

we obtain

\[ \boxed{ \frac{d}{dx}\tan x = \frac{1}{\cos^2x} = \sec^2x } \]

4. Why radians matter

These derivative rules are exact in their simple form only when angles are measured in radians.

For example,

\[ \frac{d}{dx}\sin x=\cos x \]

depends on the radian definition of angle.

Using degrees would introduce an extra constant factor.

5. Chain rule with sine

Consider

\[ y=\sin(3x). \]

The outer derivative is

\[ \cos(3x), \]

and the inner derivative is

\[ 3. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = 3\cos(3x) } \]

6. Chain rule with cosine

Consider

\[ y=\cos(x^2). \]

The outer derivative is

\[ -\sin(x^2), \]

and the inner derivative is

\[ 2x. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = -2x\sin(x^2) } \]

7. Chain rule with tangent

\[ y=\tan(5x), \]

then

\[ \frac{dy}{dx} = 5\sec^2(5x). \]

8. Product example

Consider

\[ y=x\sin x. \]

Using the product rule,

\[ \frac{dy}{dx} = 1\cdot\sin x + x\cos x. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = \sin x+x\cos x } \]

9. Second derivatives

\[ y=\sin x, \]

then

\[ y'=\cos x. \]

Differentiate again:

\[ y''=-\sin x. \]

\[ \boxed{ \frac{d^2}{dx^2}\sin x = -\sin x } \]

Similarly,

\[ \frac{d^2}{dx^2}\cos x = -\cos x. \]

10. Why this matters for differential equations

The equations

\[ y''=-y \]

have sine and cosine as solutions.

That is why trigonometric functions naturally appear in models of oscillation, waves and periodic motion.

11. Visual meaning

The derivative of sine is cosine because the slope pattern of the sine curve matches the height pattern of the cosine curve.

\[ \sin x \quad\xrightarrow{\ d/dx\ }\quad \cos x \] \[ \cos x \quad\xrightarrow{\ d/dx\ }\quad -\sin x \] \[ -\sin x \quad\xrightarrow{\ d/dx\ }\quad -\cos x \] \[ -\cos x \quad\xrightarrow{\ d/dx\ }\quad \sin x \]

Differentiating repeatedly cycles through these four functions.

12. The key rules

\[ \frac{d}{dx}\sin x=\cos x \] \[ \frac{d}{dx}\cos x=-\sin x \] \[ \frac{d}{dx}\tan x=\sec^2x \] \[ \frac{d}{dx}\sin(g(x)) = \cos(g(x))g'(x) \] \[ \frac{d}{dx}\cos(g(x)) = -\sin(g(x))g'(x) \]

These rules combine naturally with the product rule, quotient rule and chain rule.