Unizor - Derivatives - Examples - Trigonometric Functions

Notes to a video lecture on http://www.unizor.com

Derivative Examples -
Trigonometric Functions

1. f(x) = sin(x)

f'(x) = cos(x)


Proof
The function increment is
sin(x+Δx)−sin(x) =
= sin(x)·cos(Δx) +
+ cos(x)·sin(Δx) −
− sin(x) =
= sin(x)·(cos(Δx)−1) +
+ cos(x)·sin(Δx) =
= −sin(x)·2sin²(Δx/2) +
+ cos(x)·sin(Δx)
Now we can use an amazing limit
sin(x)/x→1 as x→0
Based on this,
−sin(x)·2sin²(Δx/2)/Δx → 0 and
cos(x)·sin(Δx)/Δx → cos(x)
The result for a derivative is, therefore,
f'(x) = cos(x)

2. f(x) = cos(x)

f'(x) = −sin(x)

Proof
The function increment is
cos(x+Δx)−cos(x) =
= cos(x)·cos(Δx)−sin(x)·sin(Δx) −
− cos(x) =
= cos(x)·(cos(Δx)−1) −
− sin(x)·sin(Δx) =
= −cos(x)·2sin²(Δx/2) −
− sin(x)·sin(Δx)
Now we can use an amazing limit
sin(x)/x→1 as x→0
Based on this,
−cos(x)·2sin²(Δx/2)/Δx → 0 and
sin(x)·sin(Δx)/Δx → sin(x)
The result for a derivative is, therefore,
f'(x) = −sin(x)