Guided Practice
Work through derivative problems step-by-step with scaffolded support and immediate feedback.
Learn
In this guided practice lesson, you will apply the derivative concepts and rules you learned in the previous lessons. We will walk through problems together, breaking down each step so you can build confidence before working independently.
Key Derivative Rules to Apply
- Power Rule: If f(x) = x^n, then f'(x) = n*x^(n-1)
- Constant Multiple Rule: If f(x) = c*g(x), then f'(x) = c*g'(x)
- Sum/Difference Rule: (f + g)' = f' + g'
- Product Rule: (f*g)' = f'*g + f*g'
- Quotient Rule: (f/g)' = (f'*g - f*g') / g^2
- Chain Rule: [f(g(x))]' = f'(g(x)) * g'(x)
Problem-Solving Strategy
- Identify the type of function (polynomial, rational, composite)
- Choose the appropriate rule(s)
- Apply the rule step by step
- Simplify your answer
- Check by substituting a value
Examples
Example 1: Power Rule with Multiple Terms
Find the derivative of f(x) = 3x^4 - 2x^3 + 5x - 7
Step 1: Apply the power rule to each term separately.
Step 2: For 3x^4: derivative is 3 * 4x^3 = 12x^3
Step 3: For -2x^3: derivative is -2 * 3x^2 = -6x^2
Step 4: For 5x: derivative is 5 * 1 = 5
Step 5: For -7: derivative of a constant is 0
Answer: f'(x) = 12x^3 - 6x^2 + 5
Example 2: Product Rule
Find the derivative of f(x) = (2x + 3)(x^2 - 1)
Step 1: Identify u = (2x + 3) and v = (x^2 - 1)
Step 2: Find u' = 2 and v' = 2x
Step 3: Apply product rule: f'(x) = u'v + uv'
Step 4: f'(x) = (2)(x^2 - 1) + (2x + 3)(2x)
Step 5: f'(x) = 2x^2 - 2 + 4x^2 + 6x
Answer: f'(x) = 6x^2 + 6x - 2
Example 3: Chain Rule
Find the derivative of f(x) = (3x^2 + 1)^4
Step 1: Identify outer function: u^4 and inner function: u = 3x^2 + 1
Step 2: Derivative of outer: 4u^3
Step 3: Derivative of inner: du/dx = 6x
Step 4: Apply chain rule: f'(x) = 4(3x^2 + 1)^3 * 6x
Answer: f'(x) = 24x(3x^2 + 1)^3
Practice
Try these problems on your own. Work through each step before checking the solution.
Problem 1
Find the derivative of f(x) = 5x^6 - 3x^4 + 2x^2 - 8
Show Solution
f'(x) = 30x^5 - 12x^3 + 4x
Problem 2
Find the derivative of f(x) = x^3 * sqrt(x) [Hint: rewrite sqrt(x) as x^(1/2)]
Show Solution
f(x) = x^3 * x^(1/2) = x^(7/2)
f'(x) = (7/2)x^(5/2)
Problem 3
Find the derivative of f(x) = (x^2 - 4)(x^3 + 2x)
Show Solution
Using product rule: f'(x) = 2x(x^3 + 2x) + (x^2 - 4)(3x^2 + 2)
f'(x) = 2x^4 + 4x^2 + 3x^4 + 2x^2 - 12x^2 - 8
f'(x) = 5x^4 - 6x^2 - 8
Problem 4
Find the derivative of f(x) = (2x - 5)^3
Show Solution
Using chain rule: f'(x) = 3(2x - 5)^2 * 2 = 6(2x - 5)^2
Problem 5
Find the derivative of f(x) = (x^2 + 1) / x [Hint: simplify first or use quotient rule]
Show Solution
Method 1 (simplify first): f(x) = x + 1/x = x + x^(-1)
f'(x) = 1 - x^(-2) = 1 - 1/x^2
Problem 6
Find the derivative of f(x) = sqrt(4x^2 + 9)
Show Solution
f(x) = (4x^2 + 9)^(1/2)
Using chain rule: f'(x) = (1/2)(4x^2 + 9)^(-1/2) * 8x
f'(x) = 4x / sqrt(4x^2 + 9)
Problem 7
Find the derivative of f(x) = x^2 * (x - 1)^3
Show Solution
Using product rule with chain rule:
f'(x) = 2x(x - 1)^3 + x^2 * 3(x - 1)^2
f'(x) = (x - 1)^2[2x(x - 1) + 3x^2]
f'(x) = (x - 1)^2(2x^2 - 2x + 3x^2)
f'(x) = (x - 1)^2(5x^2 - 2x)
Problem 8
Find the derivative of f(x) = 1 / (3x + 2)^2
Show Solution
f(x) = (3x + 2)^(-2)
Using chain rule: f'(x) = -2(3x + 2)^(-3) * 3
f'(x) = -6 / (3x + 2)^3
Problem 9
Find f'(2) if f(x) = x^3 - 4x^2 + 3x + 1
Show Solution
f'(x) = 3x^2 - 8x + 3
f'(2) = 3(4) - 8(2) + 3 = 12 - 16 + 3 = -1
Problem 10
Find the equation of the tangent line to f(x) = x^2 - 3x + 2 at x = 1
Show Solution
f(1) = 1 - 3 + 2 = 0, so the point is (1, 0)
f'(x) = 2x - 3, so f'(1) = 2 - 3 = -1 (slope)
Tangent line: y - 0 = -1(x - 1)
y = -x + 1
Check Your Understanding
Answer these questions to assess your mastery of derivative techniques.
- When do you need to use the chain rule instead of the power rule?
- What is the fastest way to find the derivative of (x + a)^n?
- How can you verify your derivative is correct?
- When is it easier to simplify first before differentiating?
Next Steps
- Review any problems where you made errors
- Practice additional problems from each rule category
- Move on to Word Problems to see derivatives in context
- Return to this lesson if you need a refresher on techniques