Master Derivative Rules: Sum, Product, and Chain Rule Made Simple

1 Derivative of Sum, Difference, and Product

Previously we learned basic derivatives; now we derive rules for more complex functions.

1.1 Derivative of a sum

Using the limit definition, the derivative of f+g equals f' + g'.

1.2 Derivative of a difference

The derivative of f−g is f'−g', following the same reasoning.

1.3 Derivative of a product

The product rule states (fg)' = f'g + fg'. The proof starts from the definition, rewrites the numerator, and shows the third term’s limit is zero.

2 Function composition

Given functions f: A→B and g: B→C, their composition h = g∘f maps A to C. We first apply f then g.

3 Chain rule

For a composite function h(x)=g(f(x)), the derivative is h' = g'(f(x))·f'(x).