IB Math AA SL Topic 5.3: Differentiating polynomials | Notes & Flashcards | Aimnova

IB Math AA SL — Differentiating polynomials

Topic 5.3 of IB Mathematics: Analysis and Approaches covers Differentiating polynomials, which is part of Unit 5: Calculus. Students explore key concepts including Differentiating powers, Gradient at a point. A strong understanding of differentiating polynomials is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Differentiating polynomials

Key Idea: Differentiating a polynomial finds its gradient function f'(x) — the rate of change at every point. It's a pure Paper 1 by-hand skill and the foundation for tangents, maxima and minima later in the course.

📉 The power rule

\frac{d}{dx}\big(x^{n}\big) = n\,x^{\,n-1}

$n$: the power — bring it down to multiply
$n-1$: the new power — one less than before

Multiply by the power, then subtract 1 from it. A constant multiple stays: d/dx(a·xⁿ) = a·n·xⁿ⁻¹. A standalone constant → 0 (its graph is flat). And x means x¹, so it differentiates to 1.

🔁 Rewrite BEFORE you differentiate

✏️ IB-style worked examples

IB-style question — differentiate a polynomial

Differentiate f(x) = 4x³ − 6x² + 2x − 11.

Step by step:

Apply the power rule to each term, keeping the signs.
$12x^{2},\ -12x,\ +2$
The constant −11 differentiates to 0.
$f'(x) = 12x^{2} - 12x + 2$

Final answer:

f'(x) = 12x² − 12x + 2

IB-style question — rewrite a fraction and a root first

Differentiate g(x) = 5/x² + √x.

Step by step:

Rewrite each term as a power.
$g(x) = 5x^{-2} + x^{1/2}$
Power rule: 5(−2)x⁻³ and ½x⁻¹⁄².
$g'(x) = -10x^{-3} + \tfrac{1}{2}x^{-1/2}$
Tidy back into fraction / root form.
$g'(x) = -\frac{10}{x^{3}} + \frac{1}{2\sqrt{x}}$

Final answer:

g'(x) = −10/x³ + 1/(2√x)

IB-style question — gradient at a point

Find the gradient of y = x³ − 5x at the point where x = 2.

Step by step:

Differentiate first.
$\frac{dy}{dx} = 3x^{2} - 5$
Now substitute x = 2.
$3(2)^{2} - 5 = 12 - 5 = 7$

Final answer:

The gradient at x = 2 is 7.

IB-style question — find x for a given gradient

The curve y = x² − 6x has gradient 4 at one point. Find x there.

Step by step:

Differentiate and set equal to the gradient.
$f'(x) = 2x - 6 = 4$
Solve for x.
$2x = 10 \Rightarrow x = 5$

Final answer:

x = 5.

Important: You cannot apply the power rule to √x or 1/x² as written. Rewrite them as x¹/² and x⁻² before differentiating — forgetting this is the single biggest error in this topic. (And differentiate before you substitute a value, never after.)

Tap each card to reveal the answer.

Exam Tips

Power rule: multiply by the power, then subtract 1 from it.
Differentiate a polynomial term by term; standalone constants become 0.
Rewrite fractions (1/xⁿ = x⁻ⁿ) and roots (√x = x¹/²) as powers BEFORE differentiating.
Gradient at a point = f'(a): differentiate first, then substitute.
For 'where is the gradient m?', solve f'(x) = m — a quadratic f' may give two answers. A tangent's gradient is tan(angle with the x-axis).

IB Math AA SL — Differentiating polynomials

Key concepts in Differentiating polynomials

Key Idea: Differentiating a polynomial finds its gradient function f'(x) — the rate of change at every point. It's a pure Paper 1 by-hand skill and the foundation for tangents, maxima and minima later in the course.

📉 The power rule

\frac{d}{dx}\big(x^{n}\big) = n\,x^{\,n-1}

$n$: the power — bring it down to multiply
$n-1$: the new power — one less than before

Multiply by the power, then subtract 1 from it. A constant multiple stays: d/dx(a·xⁿ) = a·n·xⁿ⁻¹. A standalone constant → 0 (its graph is flat). And x means x¹, so it differentiates to 1.

🔁 Rewrite BEFORE you differentiate

✏️ IB-style worked examples

IB-style question — differentiate a polynomial

Differentiate f(x) = 4x³ − 6x² + 2x − 11.

Step by step:

Apply the power rule to each term, keeping the signs.
$12x^{2},\ -12x,\ +2$
The constant −11 differentiates to 0.
$f'(x) = 12x^{2} - 12x + 2$

Final answer:

f'(x) = 12x² − 12x + 2

IB-style question — rewrite a fraction and a root first

Differentiate g(x) = 5/x² + √x.

Step by step:

Rewrite each term as a power.
$g(x) = 5x^{-2} + x^{1/2}$
Power rule: 5(−2)x⁻³ and ½x⁻¹⁄².
$g'(x) = -10x^{-3} + \tfrac{1}{2}x^{-1/2}$
Tidy back into fraction / root form.
$g'(x) = -\frac{10}{x^{3}} + \frac{1}{2\sqrt{x}}$

Final answer:

g'(x) = −10/x³ + 1/(2√x)

IB-style question — gradient at a point

Find the gradient of y = x³ − 5x at the point where x = 2.

Step by step:

Differentiate first.
$\frac{dy}{dx} = 3x^{2} - 5$
Now substitute x = 2.
$3(2)^{2} - 5 = 12 - 5 = 7$

Final answer:

The gradient at x = 2 is 7.

IB-style question — find x for a given gradient

The curve y = x² − 6x has gradient 4 at one point. Find x there.

Step by step:

Differentiate and set equal to the gradient.
$f'(x) = 2x - 6 = 4$
Solve for x.
$2x = 10 \Rightarrow x = 5$

Final answer:

x = 5.

Important: You cannot apply the power rule to √x or 1/x² as written. Rewrite them as x¹/² and x⁻² before differentiating — forgetting this is the single biggest error in this topic. (And differentiate before you substitute a value, never after.)

Tap each card to reveal the answer.

Exam Tips

Power rule: multiply by the power, then subtract 1 from it.
Differentiate a polynomial term by term; standalone constants become 0.
Rewrite fractions (1/xⁿ = x⁻ⁿ) and roots (√x = x¹/²) as powers BEFORE differentiating.
Gradient at a point = f'(a): differentiate first, then substitute.
For 'where is the gradient m?', solve f'(x) = m — a quadratic f' may give two answers. A tangent's gradient is tan(angle with the x-axis).

IB Math AA SL — Differentiating polynomials

Key concepts in Differentiating polynomials

📉 The power rule

🔁 Rewrite BEFORE you differentiate

✏️ IB-style worked examples

IB-style question — differentiate a polynomial

IB-style question — rewrite a fraction and a root first

IB-style question — gradient at a point

IB-style question — find x for a given gradient

Exam Tips

What you'll learn in Topic 5.3

Study resources — 5.3 Differentiating polynomials

Differentiating powers

Gradient at a point

Ready to study Differentiating polynomials?

Ready to practice?

IB Math AA SL — Differentiating polynomials

Key concepts in Differentiating polynomials

📉 The power rule

🔁 Rewrite BEFORE you differentiate

✏️ IB-style worked examples

IB-style question — differentiate a polynomial

IB-style question — rewrite a fraction and a root first

IB-style question — gradient at a point

IB-style question — find x for a given gradient

Exam Tips

What you'll learn in Topic 5.3

Study resources — 5.3 Differentiating polynomials

Differentiating powers

Gradient at a point

Ready to study Differentiating polynomials?

Ready to practice?