Types of Differentiation

I have classified differentiation into different types.

Classification by Operation

Multiplication by a Constant

$$(cf(x))'=cf'(x)$$

Addition

$$(f(x)+g(x))'=f'(x)+g'(x)$$

Subtraction

$$(f(x)-g(x))'=f'(x)-g'(x)$$

Multiplication

$$(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$$

Division

$$(\frac{g(x)}{f(x)})'=\frac{f(x)g'(x)-f'(x)g(x)}{(f(x))^2}$$

Composition (Chain Rule)

$$(f(g(x)))'=f'(g(x))·g'(x)$$

You can compute \(f'(g(x))\) as follows.

1. Compute \(f'(x)\).
2. Substitute \(x\) with \(g(x)\) in \(f'(x)\).

Various Functions

\((x^r)\) Form

If \(r\) is a real number,

$$(x^r)'=rx^{r-1}$$

Natural Logarithmic Function

$$(ln(x))'=\frac{1}{x}$$

Logarithmic Function

$$(log_{a}(x))'=\frac{1}{x·ln(a)}$$

Natural Exponential Function

$$(e^x)'=e^x$$

Exponential Function

$$(a^x)'=a^{x}·ln(a)$$