Advanced · 15 min
An introduction to derivatives
Differentiate simple polynomials and evaluate their slopes.
Apply the power rule
For positive integer n, the derivative of xⁿ is nxⁿ⁻¹. A constant has derivative zero. Multiply by any constant coefficient and differentiate sums term by term.
Worked example
If f(x) = 3x² + 2x + 7, then f′(x) = 6x + 2.
Evaluate the derivative
The derivative is a function describing instantaneous rate of change. Substitute an input to get the slope there. A negative derivative indicates a falling output locally.
Worked example
For f(x) = x², f′(x) = 2x, so the slope at x = −2 is −4.
Connect slope and motion
- For position s(t) = 2t² metres, find the velocity function by differentiation.
- Calculate the velocity at t = 3 seconds and state its units.