1. Why Differentiation Rules Matter
The basic rules of differentiation make it easy to find derivatives without going back to the limit definition every time. These rules help break down functions into manageable parts so derivatives can be computed quickly and cleanly.
2. Constant Rule
The derivative of a constant number is always 0.
2.1. Formula
\frac{d}{dx}(c) = 0
2.2. Example
\frac{d}{dx}(7) = 0
3. Power Rule
This is one of the most frequently used rules. It applies to any real power of x.
3.1. Formula
\frac{d}{dx}(x^n) = nx^{n-1}
Works for integer, fractional, and negative exponents.
3.2. Example
\frac{d}{dx}(x^5) = 5x^4
4. Constant Multiple Rule
A constant can be pulled out of the derivative.
4.1. Formula
\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)
4.2. Example
\frac{d}{dx}(4x^3) = 4 \cdot 3x^2 = 12x^2
5. Sum and Difference Rule
The derivative of a sum is the sum of derivatives. The same applies to subtraction.
5.1. Formula
\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)
5.2. Example
\frac{d}{dx}(x^2 + x^3) = 2x + 3x^2
6. Product Rule (Introduction)
The derivative of a product is not the product of the derivatives. This rule explains how to differentiate multiplied functions.
6.1. Formula
\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)
6.2. Example
\frac{d}{dx}[x^2 \cdot e^x] = x^2 e^x + 2x e^x
7. Quotient Rule (Introduction)
The quotient rule is used when one function is divided by another.
7.1. Formula
\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}
7.2. Example
\frac{d}{dx}\left(\frac{x^2}{x+1}\right) = \frac{(x+1)(2x) - x^2(1)}{(x+1)^2}
8. Chain Rule (Introduction)
The chain rule handles derivatives of composite functions — functions inside other functions.
8.1. Idea
Differentiate the outer function first, then multiply by the derivative of the inner function.
8.2. Example
\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
9. Combining Rules
Real problems often require using several rules together. Differentiation becomes easier once these basic rules are familiar, since almost any expression can be broken into smaller parts.