Maths (university) · Part 03
Differentiation
The derivative: how fast things change, and the rules for finding it.
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- DerivativeThe derivative of f at a is the limit of (f(a + h) - f(a)) / h as h tends to 0. It measures the instantaneous rate of change of f at a.
- Tangent lineThe straight line that touches a curve at one point and has the same slope there. Its equation is y - f(a) = f'(a)(x - a).
- Rate of changeThe derivative with respect to time, or any other input, tells you how fast the quantity changes. Position gives velocity, and velocity gives acceleration.
- Power ruleThe derivative of x^n is n x^(n-1). Constants disappear, and you differentiate sums term by term.
- Product ruleThe derivative of u times v is u'v + uv'. You cannot just multiply the two derivatives.
- Quotient ruleThe derivative of u over v is (u'v - uv') divided by v^2. The order of the top terms matters.
- Chain ruleTo differentiate a function inside another function, differentiate the outer one (leaving the inside alone), then multiply by the derivative of the inside.
- Standard derivativesKnow these by heart: sin x gives cos x, cos x gives -sin x, e^x gives e^x, and ln x gives 1/x for x > 0.
- Implicit differentiationWhen y is tangled up with x in an equation, differentiate both sides with respect to x and treat y as a function of x, using the chain rule.
- DifferentiabilityA function is differentiable at a if its derivative exists there. A sharp corner, a vertical tangent or a break means it is not.