CLASS 12 · FREE LEARNING CORNER

Derivatives, integrals & probability

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CLASS 12 · SAMPLE MODULE

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One idea to remember

A derivative measures a function’s rate of change. For a positive integer n, d(xⁿ)/dx = nxⁿ⁻¹. The derivative of a constant is zero; differentiate a sum term by term.

Worked example: Differentiate f(x) = x³ + 2x − 7.

The derivative of x³ is 3x².
The derivative of 2x is 2; the derivative of −7 is 0.
Therefore f′(x) = 3x² + 2. At x = 1, the rate of change is 5.

Your turn

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1. Differentiate 4x² − 3x + 5.

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8x − 3

Differentiate term by term: d(4x²)/dx = 8x, d(−3x)/dx = −3, and d(5)/dx = 0.

2. Evaluate ∫₀² x dx.

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An antiderivative is x²/2. Evaluate: (2²/2) − (0²/2) = 2.

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