Trigonometric Derivatives

Trigonometric derivatives are six essential formulas used constantly in calculus. The key pattern: $\sin x$ and $\cos x$ derivatives alternate with a sign; $\tan x$ , $\cot x$ , $\sec x$ , and $\csc x$ follow from these. Remember that "co-functions" ( $\cos$ , $\cot$ , $\csc$ ) have negative derivatives.

Trigonometric Derivatives

The six trigonometric derivatives are fundamental formulas that must be memorized—they appear repeatedly throughout calculus and cannot always be re-derived during problem-solving:

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

The Sign Pattern

A critical pattern emerges: derivatives of $\sin x$ , $\tan x$ , and $\sec x$ are positive, while derivatives of $\cos x$ , $\cot x$ , and $\csc x$ carry negative signs. The most reliable memory aid is that all "co-functions" have negative derivatives. This is where students most frequently make errors.

The relationship is visible graphically: wherever $\sin x$ reaches a peak or valley (slope = 0), its derivative $\cos x$ crosses zero, confirming the derivative correctly tracks the slope.

Graph illustrating Trigonometric Derivatives

Applying the Rules

When differentiating more complex expressions, combine these formulas with the constant multiple and sum/difference rules.

Example: Find the derivative of $f(x) = 3\sin x - 5\cos x$ : $f'(x) = 3\cos x - 5(-\sin x) = 3\cos x + 5\sin x$

Another example: For $g(x) = 4\tan x - \csc x + 7$ : $g'(x) = 4\sec^2 x - (-\csc x \cot x) = 4\sec^2 x + \csc x \cot x$

Notice how the negative sign in the $\csc x$ derivative flips when subtracted.

Common Pitfall

The most frequent mistake is confusing which functions get the minus sign. If you memorize "co-functions are negative," you'll avoid the majority of errors in this topic.

Trigonometric Derivatives