- Define and explain Introducing Differentiation in your own words
- Use key terms such as Stationary point accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
This lesson focuses on Introducing Differentiation: gradients from first principles ideas, the power rule, and stationary points.
Gradients from first principles ideas, the power rule, and stationary points.
Key ideas
Differentiating polynomials
From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule. So f(x) = 3x³ − 5x² + 2x − 7 gives f′(x) = 9x² − 10x + 2, and at x = 2 the gradient is 36 − 20 + 2 = 18.
Stationary points and nature
Solve f′(x) = 0 to find stationary points, then use the second derivative or a sign diagram: a positive second derivative means a minimum, a negative one a maximum. For f(x) = x³ − 3x, f′(x) = 3x² − 3 = 0 gives x = ±1, with a maximum at x = −1 and minimum at x = 1.
Key term — Stationary point: A point where the derivative is zero, so the tangent is horizontal: a maximum, minimum or point of inflection.
Find the stationary points of y = x² − 6x + 5.
dy/dx = 2x − 6 = 0 → x = 3; stationary point (3, −4), a minimum since d²y/dx² = 2, which is positive.
Answer: dy/dx = 2x − 6 = 0 → x = 3; stationary point (3, −4), a minimum since d²y/dx² = 2, which is positive.
- Applying the power rule to xⁿ incorrectly Multiply by the old power, then reduce it by one: the derivative of x⁵ is 5x⁴, not 5x⁵ or x⁴.
- Differentiating the constant term to 1 or x The derivative of any constant is 0 — it contributes no change.
Practice
dy/dx = 12x² + 6.
5 + 9 × 4 = 5 + 36 = 41.
2x³ + 4x + C.
[x²]₀³ = 9 − 0 = 9.
Quick check
Which of these best defines "Stationary point"?
Find f(g(x)) given f(x) = x² and g(x) = 2x + 1.
- Introducing Differentiation: gradients from first principles ideas, the power rule, and stationary points.
- Differentiating polynomials: From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule.
- Function: A rule that assigns each input exactly one output, written y = f(x); the set of allowed inputs is the domain.
- Watch out for: applying the power rule to xⁿ incorrectly