Trigonometric rule · Derivative Calculator

What it says

Everything follows from one limit, $\lim_{h \to 0}\frac{\sin h}{h} = 1$, which says that near zero the sine of an angle is the angle itself. Feeding that into the difference quotient for $\sin x$ produces $\cos x$, and the rest of the family follows by the quotient rule.

The full table: $\left(\sin x\right)' = \cos x$, $\left(\cos x\right)' = -\sin x$, $\left(\tan x\right)' = \sec^{2}x$, $\left(\sec x\right)' = \sec x\tan x$, $\left(\csc x\right)' = -\csc x\cot x$, $\left(\cot x\right)' = -\csc^{2}x$. Every co-function carries a minus sign — a reliable memory aid.

Differentiating $\sin$ four times returns to $\sin$, which is why higher derivatives of trigonometric functions cycle with period four.

When it applies

Any of the six trigonometric functions applied to the variable.
With the chain rule when the angle is a compound expression, as in $\sin(3x)$.
All of it assumes radians — in degrees an extra factor of $\frac{\pi}{180}$ appears.

Five worked examples

Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.

Example 1

\frac{d}{dx}\left[\sin\left(x\right)\right]

Straight from the table.
$\frac{d}{dx}\left[\sin x\right] = \cos x$

Answer

\cos\left(x\right)

The pair runs a quarter-turn apart. f′ peaks where f crosses the axis, the steepest part of the wave, and reads zero at the peaks of f. Drag the crosshair along and watch the two readouts trade places every quarter period.

The sine curve is steepest where the cosine peaks, at $x = 0$, and flat at its own peaks.

Open sin(x) in the calculator

Example 2

\frac{d}{dx}\left[\cos\left(x\right)\right]

The co-function picks up a minus sign.
$\frac{d}{dx}\left[\cos x\right] = -\sin x$

Answer

-\sin\left(x\right)

f starts at its peak while f′ starts at zero, then dips below the axis as the cosine falls.

Differentiating twice more gives $-\cos x$ then $\sin x$, completing the four-step cycle.

Open cos(x) in the calculator

Example 3

\frac{d}{dx}\left[\tan\left(x\right)\right]

Write $\tan x = \frac{\sin x}{\cos x}$ and use the quotient rule.
$= \frac{\cos^{2}x + \sin^{2}x}{\cos^{2}x}$
The Pythagorean identity makes the numerator $1$.
$= \frac{1}{\cos^{2}x} = \sec^{2}x$

Answer

\sec^{2}\left(x\right)

Branches climbing between the poles, with f′ a chain of U-shaped bumps dipping lowest wherever $\tan x$ crosses the axis.

Always at least $1$, which is why the tangent graph never flattens out.

Open tan(x) in the calculator

Example 4

\frac{d}{dx}\left[\sec\left(x\right)\right]

$\sec x = \left(\cos x\right)^{-1}$; chain rule with the power rule.
$= -\left(\cos x\right)^{-2}\cdot\left(-\sin x\right)$
Split into two familiar factors.
$= \frac{\sin x}{\cos^{2}x} = \sec x\tan x$

Answer

\sec\left(x\right) \tan\left(x\right)

f has U-shaped and ∩-shaped branches either side of each pole, and f′ passes through zero at the tip of each one.

The two minus signs cancel — one from the power rule, one from the derivative of $\cos$.

Open sec(x) in the calculator

Example 5

\frac{d}{dx}\left[\sin\left(x\right) \cos\left(x\right)\right]

Product rule.
$= \cos x\cos x + \sin x\left(-\sin x\right)$
Collect.
$= \cos^{2}x - \sin^{2}x = \cos 2x$

Answer

\cos^{2}\left(x\right) - \sin^{2}\left(x\right)

f is a half-height sine of double frequency, and f′ is twice as tall again.

Consistent with $\sin x\cos x = \tfrac{1}{2}\sin 2x$, whose derivative is $\cos 2x$.

Open sin(x) cos(x) in the calculator

Watch out for

Only $\cos$, $\csc$ and $\cot$ — the co-functions — produce a minus sign.
$\sin^{2}x$ means $\left(\sin x\right)^{2}$ and needs the chain rule; $\sin(x^{2})$ is a different function.
These derivatives are valid for angles in radians only.

Chain rule Inverse trigonometric rule Hyperbolic rule

What it says

When it applies

Five worked examples

Watch out for

Related rules