Why rules?
Computing every derivative from the limit definition is slow. A handful of rules cover almost everything you will meet. Learn them as patterns, then practise until they are automatic.
The core rules
- Constant: the derivative of a constant is 0.
- Power rule: the derivative of x^n is n*x^(n-1). So x^5 becomes 5x^4, and sqrt(x) = x^(1/2) becomes (1/2)x^(-1/2).
- Constant multiple and sum: pull out constants and differentiate term by term.
- Product rule: (uv)' = u'v + uv'. Differentiate one factor at a time and add.
- Quotient rule: (u/v)' = (u'v - uv')/v^2. The order on top matters, and a good mnemonic is "low d-high minus high d-low, over low squared".
- Chain rule: for a composition f(g(x)), the derivative is f'(g(x)) * g'(x). Differentiate the outer function with the inside left alone, then multiply by the derivative of the inside.
Worked example 1: product rule
Differentiate h(x) = (x^2 + 1)(x^3 - 2x). With u = x^2 + 1 and v = x^3 - 2x we have u' = 2x and v' = 3x^2 - 2. Then h'(x) = 2x(x^3 - 2x) + (x^2 + 1)(3x^2 - 2). Expanding gives 2x^4 - 4x^2 + 3x^4 + x^2 - 2, which collects to 5x^4 - 3x^2 - 2. A cross-check: multiplying first gives x^5 - x^3 - 2x, whose derivative is 5x^4 - 3x^2 - 2, the same answer.
Worked example 2: chain rule
Differentiate (4x + 1)^6. The outer function is "something to the sixth power" and the inside is 4x + 1. So the derivative is 6(4x + 1)^5 * 4 = 24(4x + 1)^5. A frequent slip is to forget the factor 4 from the inside.
Second derivatives
Differentiate twice for f''(x). If s(t) is position, then s'(t) is velocity and s''(t) is acceleration. For f(x) = x^4, f'(x) = 4x^3 and f''(x) = 12x^2.
Common mistakes
- Multiplying the derivatives of two factors instead of using the product rule.
- Reversing the numerator of the quotient rule, which flips the sign.
- Forgetting the inner derivative in the chain rule.
- Using the power rule on a variable exponent: the derivative of 2^x is not x*2^(x-1).