Study Guide
Differentiation: Product, Quotient and Chain Rules
A senior study guide to the three rules that let you differentiate products, quotients and composite functions. Each worked example names…
- Difficulty
- Level 3 · Challenge
- Time
- 30 min
Study Guide11th GradeLevel 3 · Challenge
Free 11th Grade Differentiation Basics: The Power Rule
Introduces the derivative as a slope function and sets out the power rule for differentiating polynomials, including negative and fractional powers. Worked examples cover rewriting terms before differentiating, finding the slope at a point, tangent lines and stationary points.
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- Grade
- 11th Grade
- Subject
- Math
- Topic
- Calculus
- Difficulty
- Level 3 · Challenge
- Estimated time
- 20 minutes
- Curriculum
- Texas grade levels (TEKS)
- Answers
- Not applicable
- Format
- PDF (A4) + print
Students will practice
- understanding the derivative as a slope function
- applying the power rule to polynomial terms
- rewriting roots and reciprocals as powers before differentiating
- finding slopes, tangents and stationary points
Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.
What's next?
Completed: Differentiation Basics: The Power Rule
How to use this study guide
- Read it together first, pausing at each worked example to try the step before reading the answer.
- Attempt the "Check yourself" questions at the end without looking back.
- Then practice with a worksheet from the pathway above and finish with the topic test.
Common questions
Who is this study guide for?
11th Grade students (typically ages 16–17) working on calculus. It is pitched at level 3 · challenge.
Are the answers included?
This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.
How long does it take?
About 20 minutes. Short, regular sessions work best: two or three a week beats one long one.
Do I need to sign up to download?
No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.
What should we do next?
Try 11th Grade Differentiation (Power Rule) Worksheet — Level 1.
Preview
Success TutoringWorksheets11th Grade · Mathematics · CalculusDifferentiation Basics: The Power Rule
What you need to know
The derivative of a function tells you the slope (steepness) of its graph at any point — in other words, how fast y is changing as x changes. It is written as or f′(x). For a straight line the slope is the same everywhere, but for a curve the slope changes from point to point, so the derivative is itself a function of x.
The power rule. If y = xn, then [frac dy/dx] = nx[sup n − 1]. Bring the power down to the front as a multiplier, then reduce the power by 1. A constant multiplier stays where it is, and you differentiate a sum term by term.
| Function | Derivative | Why |
|---|---|---|
| x2 | 2x | power 2 comes down, power becomes 1 |
| x3 | 3x2 | power 3 comes down, power becomes 2 |
| 4x5 | 20x4 | 4 × 5 = 20, power becomes 4 |
| x | 1 | x = x1, so 1 × x0 = 1 |
| 7 (a constant) | 0 | a constant does not change |
| x−1 | −x−2 | −1 comes down, power becomes −2 |
| x1/2 | x−1/2 | comes down, power becomes − |
Differentiate y = 3x4 − 2x2 + 7x − 5
- 3x4 → 3 × 4x3 = 12x3
- −2x2 → −2 × 2x = −4x
- 7x → 7
- −5 → 0
- = 12x[sup 3] − 4x + 7
Slope at a point: f(x) = x3 − 6x at x = 2
- f′(x) = 3x2 − 6
- f′(2) = 3 × 4 − 6 = 12 − 6 = 6
- So the curve has slope 6 at the point where x = 2.
Rewrite first: y = √x +
- Write each term as a power of x: y = x1/2 + x−2
- Differentiate: = x−1/2 − 2x−3
- Rewrite if required: = −
Expand first: y = (x + 2)(x − 3)
- Expand: y = x2 − 3x + 2x − 6 = x2 − x − 6
- = 2x − 1
Common mistake: differentiating a constant to itself instead of 0, and forgetting to multiply the existing coefficient by the power (4x5 becomes 20x4, not 4x4). Another common slip: the power rule cannot be applied to a product or quotient directly — expand or rewrite first.
Equation of the tangent to y = x2 at x = 3
- Point: when x = 3, y = 9, so the point is (3, 9).
- Slope: = 2x, so at x = 3 the slope is 6.
- Line through (3, 9) with slope 6: y − 9 = 6(x − 3)
- y = 6x − 9
A stationary point is where the slope is zero. Set f′(x) = 0 and solve for x, then substitute back into the original function to find y. For y = x2 − 4x: = 2x − 4 = 0 gives x = 2, and y = 4 − 8 = −4, so the stationary point is (2, −4).
Check yourself
- Differentiate y = x6.
- Differentiate y = 5x3 − 2x.
- If f(x) = 2x2 + 3x, find f′(1).
- Differentiate y = .
- Differentiate y = (x + 1)2.
- Find the stationary point of y = x2 − 4x.
Answers: 1. 6x5 2. 15x2 − 2 3. f′(x) = 4x + 3, so f′(1) = 7 4. y = x−1, so = −x−2 = − 5. Expand to x2 + 2x + 1, so = 2x + 2 6. (2, −4)
Preview — the PDF contains the full resource.
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- Last reviewed
- 1 October 2026
- How it was made
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