A derivative is a slope detector. Hand it any curve and it tells you how steep that curve is at every single point. Four rules — and you can differentiate almost anything.
Most derivatives can be computed with four rules. Memorise these and you can differentiate almost any reasonable function.
Optimisation, motion, machine learning gradients, signal denoising, financial modelling — every place we ask 'how does this change?' is asking for a derivative.
The four rules
- Power: d/dx(xⁿ) = n·xⁿ⁻¹
- Sum: (f + g)' = f' + g'
- Product: (f·g)' = f'·g + f·g'
- Chain: (f(g(x)))' = f'(g(x)) · g'(x)
Quotient rule is just product + chain combined: (f/g)' = (f'g − fg') / g². But honestly, most people prefer to rewrite f/g as f · g⁻¹ and use product + chain.
d/dx(3x⁴)
Power rule: 3 · 4x³ = 12x³.
d/dx(sin(x²))
Chain rule: cos(x²) · 2x = 2x·cos(x²).
Differentiate f(x) = (x² + 1)⁵.
The product rule is not f'g'. Many students try (f·g)' = f'·g'. It isn't — you need both terms: f'g + fg'. Always write both before simplifying.
- Power, sum, product, chain — four rules cover almost everything.
- Chain rule when one function is inside another. Differentiate outer × derivative of inner.
- When stuck, simplify before differentiating — sometimes algebra dodges a hairy product.