／var／log marcus chiu

❯

❯

Mathematical Analysis

❯

❯

Calculus Foundational Knowledge

❯

Explicit Differentiation／Derivative／Differentiable／Differential Functions (differentiable at a point - differentiable over an interval - differentiable everywhere／over the entire domain)

❯

Differential／Derivative Calculus - Examples

Derivative of Exponential Function (a^x and e^x)

Created on Nov 19, 2021 · Last Modified on Aug 28, 2023

Derivative of Exponential Function

derivative of 𝑎^𝑥 is 𝑎^𝑥·𝑙𝑛(𝑎), where 𝑎 is a constant
derivative of 𝑒^𝑥 is 𝑒^𝑥·𝑙𝑛(𝑒)=𝑒^𝑥, where 𝑒 is Euler’s number

Intuition

$f (x) = 2^{x}$
$f^{'} (x) = l i m_{d x \to 0} \frac{f ( x + d x ) - f ( x )}{d x}$
$f^{'} (x) = l i m_{d x \to 0} \frac{2 ^{x + d x} - 2 ^{x}}{d x}$
$f^{'} (x) = l i m_{d x \to 0} \frac{2 ^{x} 2 ^{d x} - 2 ^{x}}{d x}$
$f^{'} (x) = 2^{x} \cdot l i m_{d x \to 0} \frac{2 ^{d x} - 1}{d x}$
$f^{'} (x) = 2^{x} \cdot 0.69314718$
$f^{'} (x) = 2^{x} \cdot l n (2)$