Derivatives of sin and cos: formula, calculator and worked example

What the symbols mean

$D$: the value of $f′(x)$ at the chosen x
$A$: the coefficient of $sin (kx)$
$B$: the coefficient of $cos (kx)$
$k$: the number multiplying x inside (angular frequency)
$x$: the point, in radians

When to use it

Use it for anything that oscillates (springs, waves, alternating current, tides) and for finding the maxima and minima of trig expressions. The calculator differentiates $Asin (kx) + Bcos (kx)$ . Always work in radians: in degrees every derivative picks up an extra factor $π/180$ .

Worked example

Find the derivative of f(x) = 2 sin(3x) at x = 1 radian (A = 2, B = 0, k = 3).

For $f (x) = 2 \sin (3 x)$ , differentiate each part: sin(kx) becomes k cos(kx) and cos(kx) becomes −k sin(kx) (chain rule).
$f' (x) = A k \cos (k x) - B k \sin (k x)$
Substitute and work it out (x in radians):
$D = 2 \cdot 3 \cos (3 \cdot 1) - 0 \cdot 3 \sin (3 \cdot 1)$
$= 6 \cos (3) - 0 \sin (3)$
$\approx 6 \cdot (- 0.989992) - 0 \cdot 0.14112$
$\approx 6 \cdot (- 0.989992) - 0$
$\approx - 5.93995$
Angles are in radians: the rule d/dx sin x = cos x is only true in radians (in degrees an extra factor π/180 appears).

Answer f′(1) ≈ −5.940 (that is 2 × 3 × cos 3)

Why it works

The slope of the sine curve is largest at 0, zero at the peaks and negative on the way down: exactly the pattern of cosine. Formally it follows from $lim[h → 0] (sin h)/h = 1$ , which is only true when h is measured in radians.

Common mistakes

Using degrees: $cos (3^°)$ is not $cos 3$ . Calculus with trig needs radian mode.
Losing the minus sign: the derivative of cos is $−sin$ .
Forgetting the chain-rule factor k: $d/(dx)sin (3x) = 3cos (3x)$ .

Derivatives of sin and cos

Calculator

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What the symbols mean

When to use it

Worked example

Why it works

Common mistakes