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

__Derivative Examples -__

Exponential Functions

Exponential Functions

1.

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

*f'(x) = e*^{x}*Proof*

Function increment is

*e*^{x+Δx}−

**=**

*e*^{x}=

**·(**

*e*^{x}

*e*^{Δx}−

**)**

*1*Therefore, to find derivative, we have to find a limit of the following expression as Δ

**:**

*x→0***·(**

*e*^{x}

*e*^{Δx}−

**)**

*1***Δ**

*/*

*x*Since

(

*e*^{Δx}−

**)**

*1***Δ**

*/***→**

*x***as Δ**

*1***,**

*x → 0*the derivative is

*f'(x) = e*^{x}2.

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

*f'(x) = ln(a)·a*^{x}*Proof*

We will use the identity

*a = e*^{ln(a)}that follows directly from the definition of the natural logarithm.

The function increment looks now as follows:

*a*^{x+Δx}−

**=**

*a*^{x}=

**·(**

*a*^{x}

*e*^{Δx·ln(a)}−

**)**

*1*Therefore, to find derivative, we have to find a limit of the following expression as Δ

**:**

*x→0***·(**

*a*^{x}

*e*^{Δx·ln(a)}−

**)**

*1***Δ**

*/*

*x*Obviously,

(

*e*^{Δx·ln(a)}−

**)**

*1***(Δ**

*/***) →**

*x·ln(a)*

*1*Therefore,

(

*e*^{Δx·ln(a)}−

**)**

*1***Δ**

*/***→**

*x*

*ln(a)*from which follows that the derivative equals to

*f'(x) = ln(a)·a*^{x}
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