GCSE Further Maths · Topic 3
Calculus
Calculus is the mathematics of change.
We will differentiate powers of x to find gradients, tangents, normals and stationary points.
Then we integrate to find areas, and use both in kinematics.
- 5 subtopics
- 223 questions
- Free: 3.1 notes and 10 questions
In this topic
- 3.1Differentiation: the gradient function of powers of xFree below · 22 questions
- 3.2Tangents and normals to curvesOpen in Full Marks · 52 questions
- 3.3Stationary points: maxima, minima and curve sketchingOpen in Full Marks · 55 questions
- 3.4Integration: indefinite and definite integrals and the area under a curveOpen in Full Marks · 47 questions
- 3.5Kinematics with calculus: displacement, velocity and accelerationOpen in Full Marks · 47 questions
3.1 · Free notes
Differentiation: the gradient function of powers of x
Key points
- The derivative dy/dx (or f′(x)) gives the gradient of a curve at any point; it measures the rate of change of y with respect to x.
- Rule: if y = axⁿ then dy/dx = naxⁿ⁻¹. Constants differentiate to 0, and you can differentiate term by term.
- Rewrite roots and fractions as powers first: √x = x^(1/2), 1/x² = x⁻². For example y = 4/x → y = 4x⁻¹ → dy/dx = −4x⁻² = −4/x².
- To find the gradient at a point, substitute the x-coordinate into dy/dx.
- The second derivative d²y/dx² is the derivative of dy/dx.
In more detail
- Worked example: y = (x² + 1)/x = x + x⁻¹, so dy/dx = 1 − x⁻² = 1 − 1/x².
- Meaning: if y is distance and x is time, dy/dx is velocity: the rate of change.
- Exam tip: always rewrite in index form and expand brackets before differentiating; you cannot differentiate a product term by term.
Practice
Differentiation: the gradient function of powers of x: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. Differentiate y = x⁵.
- 5x⁴
- x⁴
- 5x⁵
- 4x⁵
Show answer
A: 5x⁴. Multiply by the power and reduce it by 1.
2. Differentiate y = 3x² − 4x + 7.
- 3x − 4
- 6x + 7
- 6x − 4
- 6x² − 4
Show answer
C: 6x − 4. 3 × 2x − 4 + 0.
3. What is the derivative of a constant?
- 1
- 0
- The constant
- x
Show answer
B: 0. A constant has zero gradient.
4. Differentiate y = √x.
- 2√x
- √x/2
- 1/√x
- 1/(2√x)
Show answer
D: 1/(2√x). x^(1/2) → ½x^(−1/2).
5. Differentiate y = 6/x.
- −6/x²
- 6/x²
- 6
- −6/x
Show answer
A: −6/x². 6x⁻¹ → −6x⁻².
6. Find the gradient of y = x³ − 2x at x = 2.
- 4
- 12
- 10
- 6
Show answer
C: 10. dy/dx = 3x² − 2 = 12 − 2 = 10.
7. What is d²y/dx² for y = x⁴?
- 4x³
- 12x²
- 12x³
- 24x
Show answer
B: 12x². dy/dx = 4x³, then 12x².
8. Differentiate y = (x + 3)².
- 2x + 3
- x + 6
- 2(x + 9)
- 2x + 6
Show answer
D: 2x + 6. Expand: x² + 6x + 9, then differentiate.
9. At which x is the gradient of y = x² − 6x zero?
- x = 3
- x = 6
- x = 0
- x = −3
Show answer
A: x = 3. 2x − 6 = 0.
10. Differentiate y = x^(3/2).
- (2/3)x^(5/2)
- x^(1/2)
- (3/2)x^(1/2)
- (3/2)x^(3/2)
Show answer
C: (3/2)x^(1/2). Multiply by 3/2 and subtract 1 from the power.
Keep going in Full Marks
Full Marks has 22 questions on Differentiation: the gradient function of powers of x (12 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Calculus, plus games, timed exams and past papers.