Basic Differentiation Rules
Ever wonder how mathematicians quickly find slopes of complex curves? The answer lies in differentiation rules! Instead of using the limit definition every time, these shortcuts make calculus much more manageable.
The Constant Rule states that the derivative of any constant number is zero. Think about it: a constant function is just a horizontal line with no slope! The Power Rule is your go-to formula for expressions with variables raised to powers: if f = x^n, then f' = nx^. This works even with negative and fractional exponents.
When working with constants multiplied by functions, use the Constant Multiple Rule: if f = c·g, then f' = c·g'. This simply means you can pull constants outside the derivative. For example, the derivative of 3x² is 6x, because you multiply the constant (3) by the derivative of x² (which is 2x).
Pro Tip: When dealing with fractional exponents, convert them to their simplest form first. For example, √x is the same as x^, and ∛x is x^.
The Sum and Difference Rule lets you take derivatives term by term: the derivative of f ± g equals f' ± g'. This means you can differentiate each part separately, then combine the results with the same operations.
