Limits and derivatives as the start of calculus
A limit describes the value a function approaches as x gets closer to a point, even when the function is not defined at that point. Substituting x = 2 into (x² − 4)/(x − 2) gives the indeterminate form 0/0, but factorising the expression to x + 2 shows that the limit is 4. Standard limits include sin x / x → 1 and (eˣ − 1)/x → 1 as x → 0, and (xⁿ − aⁿ)/(x − a) → naⁿ⁻¹ as x → a.
The derivative measures the instantaneous rate of change of a function, which is the slope of its tangent. From first principles, f′(x) is the limit of [f(x + h) − f(x)] ÷ h as h → 0. Applying this gives d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x and d/dx(cos x) = −sin x. The product rule (uv)′ = u′v + uv′ and the quotient rule (u/v)′ = (u′v − uv′)/v² handle combinations of functions, and the chain rule handles one function inside another. For example, the derivative of x³ + 5x is 3x² + 5.
Why it matters
Derivatives describe how fast things change: velocity is the rate of change of position, marginal cost is the rate of change of total cost, and the growth rate of a population is a derivative. In Class 12 they are used to find maxima and minima, tangents and rates of change, and they lead directly to integration. Calculus is essential for physics, engineering and economics.
Key terms
- Limit
- The value a function gets closer to as its input approaches a given point.
- Indeterminate form
- An expression such as 0/0 whose value cannot be found by direct substitution.
- Derivative
- The instantaneous rate of change of a function, equal to the slope of its tangent.
- Chain rule
- The rule for differentiating a composite function by multiplying the derivatives of its parts.