Derivatives of polynomials and slope of a tangent: Class 12 practice
The derivative f′(x) gives the slope of the tangent to y = f(x). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Laws of exponents
- Straight-line equation
The idea behind derivatives of polynomials and slope of a tangent
The derivative f′(x) gives the slope of the tangent to y = f(x). For powers, d/dx(xⁿ) = nxⁿ⁻¹, and derivatives of sums are sums of derivatives.
Step-by-step method
- Differentiate term by term with the power rule.
- Substitute the x-value for the slope.
- Tangent line: y − y₀ = f′(x₀)(x − x₀).
Worked example 1 (Basic)
If f(x) = 3x³ + 1x² + 0x + 2, find f′(-3).
- f′(x) = 9x² + 2x + 0
- f′(-3) = 75
- Answer: 75
Worked example 2 (Standard)
If f(x) = 1x³ + 0x² − 3x + 9, find f′(1).
- f′(x) = 3x² + 0x − 3
- f′(1) = 0
- Answer: 0
Worked example 3 (Challenge)
Find the equation of the tangent to y = 4x³ + 3x² − 8x − 1 at x = 3.
- y′ = 12x² + 6x − 8
- Slope at x = 3: 118
- Point: (3, 110)
- y − 110 = 118(x − 3)
- Answer: y = 118x − 244
Common mistakes and misconceptions
- Forgetting that a constant's derivative is 0
- Substituting before differentiating
Practice questions
- (Basic) If f(x) = 2x³ − 2x² + 8x − 9, find f′(-3).
- (Basic) If f(x) = 3x³ + 2x² − 3x + 0, find f′(0).
- (Standard) If f(x) = 1x³ + 0x² + 8x − 9, find f′(2).
- (Standard) If f(x) = 4x³ − 6x² − 8x + 9, find f′(-1).
- (Challenge) Find the equation of the tangent to y = 4x³ − 1x² − 2x + 5 at x = 2.
- (Challenge) Find the equation of the tangent to y = 1x³ − 5x² + 7x − 4 at x = -1.
Answer key with full working
- Q1: f′(x) = 6x² − 4x + 8 → f′(-3) = 74 → Answer: 74
- Q2: f′(x) = 9x² + 4x − 3 → f′(0) = -3 → Answer: -3
- Q3: f′(x) = 3x² + 0x + 8 → f′(2) = 20 → Answer: 20
- Q4: f′(x) = 12x² − 12x − 8 → f′(-1) = 16 → Answer: 16
- Q5: y′ = 12x² − 2x − 2 → Slope at x = 2: 42 → Point: (2, 29) → y − 29 = 42(x − 2) → Answer: y = 42x − 55
- Q6: y′ = 3x² − 10x + 7 → Slope at x = -1: 20 → Point: (-1, -17) → y − -17 = 20(x − -1) → Answer: y = 20x + 3
Syllabus note
This skill is listed under "Continuity and Differentiability" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is derivatives of polynomials and slope of a tangent in?
"Continuity and Differentiability", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.