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

__Derivative Examples -__

Power Functions

Power Functions

1.

*f(x) = x*^{n}

*f'(x) = n·x*^{n−1}*Proof*

Using Newton's binomial formula,

(

**+Δ**

*x***)**

*x*^{n}=

*x*^{n}+

**·**

*n*

*x*^{n−1}·Δ

**+**

*x***·Δ**

*P*^{2}

*x*where

**is some polynomial of**

*P***and Δ**

*x***.**

*x*Therefore,

[(

**+Δ**

*x***)**

*x*^{n}−

*x*^{n}]/Δ

**=**

*x*=

**·**

*n*

*x*^{n−1}+

**·Δ**

*P*

*x*As Δ

**→**

*x***, this expression tends to**

*0***·**

*n*

*x*^{n−1}

2.

**(defined for**

*f(x) = √x***)**

*x ≥ 0***(**

*f'(x) = 1/**)*

**2√x**Notice that derivative does not exist for

**. So, the domain of a derivative is narrower than the domain of the original function.**

*x=0**Proof*

We assume that

**takes only positive values.**

*x*Let's multiply and divide the expression for function increment √

**+Δ**

*x***− √**

*x*

*x*by √

**+Δ**

*x***+ √**

*x*

*x*Since

*(a−b)·(a+b)=a²−b²*, the result will be

Δ

**/(√**

*x***+Δ**

*x***+ √**

*x***)**

*x*Dividing this by increment of argument Δ

**and going to the limit as Δ**

*x***→**

*x***, the result will be 1/(**

*0**)*

**2√x**
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