Power, Product & Quotient Rule Worksheet | DerivativeCalculus.com

📚 The Three Fundamental Rules

⚡ Power Rule

d/dx [xⁿ] = n·xⁿ⁻¹

Example:

d/dx [x⁵] = 5x⁴

d/dx [3x²] = 6x

d/dx [√x] = d/dx [x¹⸍²] = (1/2)x⁻¹⸍²

✖️ Product Rule

(f·g)' = f'·g + f·g'

Mnemonic:

"Derivative of First times Second,

plus First times Derivative of Second"

Example: (x²·sin x)' = 2x·sin x + x²·cos x

➗ Quotient Rule

(f/g)' = (f'·g - f·g')/g²

Mnemonic:

"Low D-High minus High D-Low,

Square the Bottom and Away We Go!"

Example: (sin x/x)' = (x·cos x - sin x)/x²

Problem 1 Power Rule

f(x) = 3x⁴ - 2x² + 5x - 7

1 Apply power rule to each term:

2 For 3x⁴: derivative = 3·4x³ = 12x³

3 For -2x²: derivative = -2·2x = -4x

4 For 5x: derivative = 5·1x⁰ = 5

5 For -7 (constant): derivative = 0

6 Combine results:

f'(x) = 12x³ - 4x + 5

f'(x) = 12x³ - 4x + 5

Problem 2 Product Rule

g(x) = (x² + 1)(x³ - 2x)

1 Identify f and g:

f(x) = x² + 1, g(x) = x³ - 2x

2 Find derivatives:

f'(x) = 2x, g'(x) = 3x² - 2

3 Apply product rule: (f·g)' = f'·g + f·g'

4 Compute f'·g:

2x\cdot(x³ - 2x) = 2x⁴ - 4x²

5 Compute f·g':

(x² + 1)\cdot(3x² - 2) = 3x⁴ - 2x² + 3x² - 2 = 3x⁴ + x² - 2

6 Add results:

(2x⁴ - 4x²) + (3x⁴ + x² - 2) = 5x⁴ - 3x² - 2

7 Simplify:

g'(x) = 5x⁴ - 3x² - 2

g'(x) = 5x⁴ - 3x² - 2

Problem 3 Quotient Rule

h(x) = (x² + 1)/(x - 3)

1 Identify f and g:

f(x) = x² + 1, g(x) = x - 3

2 Find derivatives:

f'(x) = 2x, g'(x) = 1

3 Apply quotient rule: (f/g)' = (f'·g - f·g')/g²

4 Compute numerator: f'·g - f·g'

5 f'·g = 2x·(x - 3) = 2x² - 6x

6 f·g' = (x² + 1)·1 = x² + 1

7 Numerator: (2x² - 6x) - (x² + 1) = x² - 6x - 1

8 Denominator: g² = (x - 3)²

9 Combine:

h'(x) = (x² - 6x - 1)/(x - 3)²

h'(x) = (x² - 6x - 1)/(x - 3)²

Problem 4 Combined Rules

y = x³·sin(x)

1 This is a product: f(x)·g(x) where:

f(x) = x³, g(x) = sin(x)

2 Find derivatives using power rule and trig derivative:

f'(x) = 3x², g'(x) = cos(x)

3 Apply product rule: (f·g)' = f'·g + f·g'

4 Compute f'·g:

3x²\cdotsin(x)

5 Compute f·g':

x³\cdotcos(x)

6 Add results:

y' = 3x²\cdotsin(x) + x³\cdotcos(x)

y' = 3x²·sin(x) + x³·cos(x)

Problem 5 Quotient Rule

f(x) = (2x³ - 5)/(x²)

1 Alternative: Simplify first using algebra:

f(x) = (2x³ - 5)/(x²) = 2x - 5x⁻²

2 Now use power rule:

f'(x) = 2 - 5\cdot(-2)x⁻³ = 2 + 10x⁻³

3 Rewrite with positive exponent:

f'(x) = 2 + 10/x³

4 Common denominator:

f'(x) = (2x³ + 10)/x³

f'(x) = (2x³ + 10)/x³ or 2 + 10/x³

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