Derivatives of Functions

Derivatives of Trig Functions

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

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)=2sin⁡x+5cos⁡xf(x)= 2\sin x+5\cos x

Step 2 — Differentiate sine term

The sine term becomes 2 cosine x.

ddx2sin⁡x=2cos⁡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.

ddx5cos⁡x=−5sin⁡x\frac{d}{dx} \hl{5} \cos x=- \hl{5} \sin x

Step 4 — Result

Combine the trig derivatives.

f′(x)=2cos⁡x−5sin⁡xf'(x)= \hlmath{2\cos x-5\sin x}
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.