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

__Derivative Examples -__

Trigonometric Functions

Trigonometric Functions

1.

*f(x) = sin(x)*

*f'(x) = cos(x)**Proof*

The function increment is

**Δ**

*sin(x+***=**

*x)−sin(x)*=

**Δ**

*sin(x)·cos(***Δ**

*x) +*

+ cos(x)·sin(+ cos(x)·sin(

**Δ**

*x) −*

− sin(x) =

= sin(x)·(cos(− sin(x) =

= sin(x)·(cos(

**Δ**

*x)−1) +*

+ cos(x)·sin(+ cos(x)·sin(

**Δ**

*x) =*

= −sin(x)·2sin²(= −sin(x)·2sin²(

**Δ**

*x/2) +*

+ cos(x)·sin(+ cos(x)·sin(

*x)*Now we can use an amazing limit

**→**

*sin(x)/x***as**

*1*

*x→0*Based on this,

**Δ**

*−sin(x)·2sin²(***Δ**

*x/2)/***and**

*x → 0***Δ**

*cos(x)·sin(***/Δ**

*x)*

*x → 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

**Δ**

*cos(x+***=**

*x)−cos(x)*=

**Δ**

*cos(x)·cos(***Δ**

*x)−sin(x)·sin(***Δ**

*x) −*

− cos(x) =

= cos(x)·(cos(− cos(x) =

= cos(x)·(cos(

**Δ**

*x)−1) −*

− sin(x)·sin(− sin(x)·sin(

**Δ**

*x) =*

= −cos(x)·2sin²(= −cos(x)·2sin²(

**Δ**

*x/2) −*

− sin(x)·sin(− sin(x)·sin(

*x)*Now we can use an amazing limit

**→**

*sin(x)/x***as**

*1*

*x→0*Based on this,

**Δ**

*−cos(x)·2sin²(***Δ**

*x/2)/***and**

*x → 0***Δ**

*sin(x)·sin(***/Δ**

*x)*

*x → sin(x)*The result for a derivative is, therefore,

*f'(x) = −sin(x)*
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