Eroxl's Notes
Differentiation Rules

Differentiation rules describe simple rules that can be used to compute a derivative of a function without using the formal limit as shown below.

General Rules

Specific Functions

Trigonometric Functions

Function Derivative Example
$$\sin{x}$$ $$\cos{x}$$ $$\deriv{}{x}\sin{x^2}=\cos{x^2} \cdot 2x$$
$$\cos{x}$$ $$-\sin{x}$$ $$\deriv{}{x}\cos{x^2}=-\sin{x^2} \cdot 2x$$
$$\tan{x}$$ $$\sec^2{x}$$ $$\deriv{}{x}\tan{x^2}=\sec^2{x^2} \cdot 2x$$

Reciprocals

Function Derivative Example
$$\csc{}x$$ $$-\csc{}x\cot{}x$$ $$\deriv{}{x}\csc{x^2} = -\csc{x^2}\cot{x^2} \cdot 2x$$
$$\sec{x}$$ $$\sec{x}\tan{x}$$ $$\deriv{}{x}\sec{x^2} = \sec{x^2}\tan{x^2} \cdot 2x$$
$$\cot{x}$$ $$-\csc^2{x}$$ $$\deriv{}{x}\cot{x^2} = -\csc^2{x^2} \cdot 2x$$

Inverses

Function Derivative Example
$$\sin^{-1}{x}$$ $$\frac{1}{\sqrt{1-x^2}}$$ $$\deriv{}{x}\sin^{-1}{x^2} = \frac{1}{\sqrt{1-x^4}} \cdot 2x$$
$$\cos^{-1}{x}$$ $$\frac{-1}{\sqrt{1-x^2}}$$ $$\deriv{}{x}\cos^{-1}{x^2} = \frac{-1}{\sqrt{1-x^4}} \cdot 2x$$
$$\tan^{-1}{x}$$ $$\frac{1}{1+x^2}$$ $$\deriv{}{x}\tan^{-1}{x^2} = \frac{1}{1+x^4} \cdot 2x$$
$$\csc^{-1}{x}$$ $$\frac{-1}{\mid x\mid\sqrt{x^2-1}}$$ $$\deriv{}{x}\csc^{-1}{x^2} = \frac{-1}{\mid x^2 \mid\sqrt{x^4-1}} \cdot 2x$$
$$\csc^{-1}{x}$$ $$\frac{-1}{\mid x \mid \sqrt{x^2-1}}$$ $$\deriv{}{x}\csc^{-1}{x^2} = \frac{-1}{\mid x^2 \mid \sqrt{x^4-1}} \cdot 2x$$
$$\cot^{-1}{x}$$ $$-\frac{-1}{1+x^2}$$ $$\deriv{}{x}\cot^{-1}{x^2} = \frac{-1}{1+x^4} \cdot 2x$$

Logarithms and Exponents

Function Derivative Example
$$\ln{x}$$ $$\frac{1}{x}$$ $$\deriv{}{x}\ln{x^2}=\frac{1}{x^2} \cdot 2x$$
$$\ln(-x)$$ $$\frac{1}{x}$$ $$\deriv{}{x}\ln(-x^2)=\frac{1}{x^2} \cdot 2x$$
$$\log_{a}{x}$$ $$\frac{1}{x\ln{a}}$$ $$\deriv{}{x}\log_{a}{x^2}=\frac{1}{x^2\ln{a}} \cdot 2x$$
$$e^{x}$$ $$e^{x}$$ $$\deriv{}{x}e^{x^2}=e^{x^2} \cdot 2x$$
$$a^{x}$$ $$a^{x}\ln{a}$$ $$\deriv{}{x}4^{x^2}=4^{x^2}\ln{4} \cdot 2x$$