Derivatives of Trig Functions - Calculus Step by Step

Differentiate f(x)=2sin(x)+5cos(x) using d/dx[sin]=cos and d/dx[cos]=-sin.

trig-derivatives The basic trig derivatives are d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = -sin(x). Combined with the sum and constant multiple rules, any linear combination of sine and cosine can be differentiated term by term.

Example

Use the basic trig derivative rules, including the negative cosine derivative.

highlighted = computed this step

Step 1 — Set up

Start with the trig sum.

f(x)= 2\sin x+5\cos x

Step 2 — Differentiate sine term

The sine term becomes 2 cosine x.

\frac{d}{dx} \hl{2} \sin x= \hl{2} \cos x

Step 3 — Differentiate cosine term

The cosine term becomes negative 5 sine x.

\frac{d}{dx} \hl{5} \cos x=- \hl{5} \sin x

Step 4 — Result

Combine the trig derivatives.

f'(x)= \hlmath{2\cos x-5\sin x}