Standard derivatives are pre-calculated differentiation results for common functions that students must memorize and apply directly without deriving them each time.
## Core concept
A derivative measures the rate of change of a function. Standard derivatives are the instantaneous rate of change formulas for elementary functions (powers, exponentials, logarithms, trigonometric, inverse trigonometric). In CA Foundation exams, you are expected to know these formulas and apply them to solve practical optimization and rate problems—not derive them from first principles.
The standard derivative formula is: if y = f(x), then dy/dx represents the derivative.
## Formula / rule
Algebraic functions: - d/dx(xⁿ) = nxⁿ⁻¹ - d/dx(1/x) = d/dx(x⁻¹) = −x⁻² = −1/x² - d/dx(√x) = d/dx(x^(1/2)) = (1/2)x^(−1/2) = 1/(2√x)
Exponential & logarithmic: - d/dx(eˣ) = eˣ - d/dx(aˣ) = aˣ ln(a) - d/dx(ln x) = 1/x - d/dx(log_a x) = 1/(x ln a)
Trigonometric: - d/dx(sin x) = cos x - d/dx(cos x) = −sin x - d/dx(tan x) = sec² x - d/dx(cot x) = −cosec² x - d/dx(sec x) = sec x tan x - d/dx(cosec x) = −cosec x cot x
Inverse trigonometric: - d/dx(sin⁻¹ x) = 1/√(1 − x²) - d/dx(cos⁻¹ x) = −1/√(1 − x²) - d/dx(tan⁻¹ x) = 1/(1 + x²)
Constants & composite: - d/dx(constant) = 0 - d/dx(x) = 1
## Common exam applications
- Optimization problems: Find maximum/minimum profit, cost, or revenue by setting dy/dx = 0.
- Rate of change: If cost C = 50x + 100 and output x changes, find dC/dx to determine marginal cost.
- Velocity & acceleration: If displacement s(t) = 5t², then velocity ds/dt = 10t.
- Elasticity of demand: E = (dQ/dP) × (P/Q); requires standard derivative of demand function.
## Common mistakes
- Forgetting the chain rule: d/dx(sin 2x) ≠ cos 2x; it equals 2cos 2x. (Standard derivative applies to innermost function; multiply by derivative of inner.)
- Confusing log and ln: ln x means natural log (base e); log_a x requires the 1/(x ln a) formula. Many students apply 1/x to all logarithms.
- Sign errors with trigonometric: d/dx(cos x) = −sin x (negative). Common reversal error.
- Incorrect inverse trig domain: sin⁻¹ x derivative is 1/√(1 − x²), valid only for |x| < 1. Exam questions test boundary awareness.
## Worked example
Problem: A firm's revenue R = 120x − 2x², where x is quantity sold. Find the marginal revenue (dR/dx) and the quantity at which marginal revenue is zero.
Solution: - dR/dx = d/dx(120x − 2x²) - Using standard rule d/dx(xⁿ) = nxⁿ⁻¹: - dR/dx = 120(1) − 2(2x) = 120 − 4x - Set dR/dx = 0: 120 − 4x = 0 → x = 30 units - At x = 30, marginal revenue is zero; beyond this, MR becomes negative (declining revenue).
This is typical CA Foundation application: use standard derivatives to find critical points for business decisions.