Grade 12 · Mathematics
Differentiate a polynomial with the power rule
A derivative describes the instantaneous rate of change of a function. For a graph y = f(x), f′(x) gives the tangent gradient at input x. Grade 12 polynomial differentiation uses the power rule: multiply by the exponent, then reduce that exponent by one. Constants differentiate to zero. Apply the rule to each term before substituting a point.
Original explanation prepared with AI assistance.
Lees hierdie verduideliking in AfrikaansWhat you will practise
Differentiate a polynomial and calculate a tangent gradient.
The method
For a term axⁿ, its derivative is anxⁿ⁻¹. Keep coefficient signs and differentiate terms separately.
Worked example
Find f′(x) and f′(2) if f(x) = 3x³ − 2x² + 5x − 7.
- Differentiate 3x³ to get 9x².
- Differentiate −2x² to get −4x, 5x to get 5 and −7 to get 0.
- So f′(x) = 9x² − 4x + 5.
- At x = 2 the gradient is 9(4) − 8 + 5 = 33.
Common mistake
The derivative of a constant is zero, not the constant itself.
Try it yourself
Differentiate h(x) = 2x⁴ − 3x + 6.
Show the explained solution
h′(x) = 8x³ − 3. Each power drops by one; the constant 6 disappears.
Curriculum and sources
Use the official CAPS subject document and your teacher’s annual plan to confirm the sequence and scope for your class. The link below is an official reference, not evidence of DBE approval of this explanation.
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