Overview
This lecture covers how to simplify derivatives using the product rule and chain rule, with step-by-step worked examples.
Product Rule and Chain Rule Overview
- The product rule: derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x).
- The chain rule: differentiate the outside function, multiply by the derivative of the inside function.
Example 1: Derivative of (x^3(2x-5)^4)
- Let (f(x) = x^3), (g(x) = (2x-5)^4).
- (f'(x) = 3x^2).
- (g'(x)): Use chain rule—4(2x-5)(^3) × 2 = 8(2x-5)(^3).
- Apply product rule: (3x^2(2x-5)^4 + x^3 \times 8(2x-5)^3).
- Factor out GCF: (x^2(2x-5)^3).
- Simplified form: (x^2(2x-5)^3[3(2x-5) + 8x]).
- Distribute and combine: Final answer is (x^2(2x-5)^3(14x-15)).
Example 2: Derivative of (x^2\sqrt{4-9x})
- Rewrite ( \sqrt{4-9x} ) as ( (4-9x)^{1/2} ).
- (f(x) = x^2), (g(x) = (4-9x)^{1/2}).
- (f'(x) = 2x).
- (g'(x)) (chain rule): ((1/2)(4-9x)^{-1/2} \times (-9) = -9/2(4-9x)^{-1/2}).
- Apply product rule: (2x(4-9x)^{1/2} + x^2 \times [-9/2(4-9x)^{-1/2}]).
- Factor out (x(4-9x)^{-1/2}).
- Combine terms to get (x[8 - (45/2)x] / \sqrt{4-9x}).
- Answer may be left as is, or denominator can be rationalized if required.
Key Terms & Definitions
- Product Rule — Rule for differentiating products: ( (fg)' = f'g + fg' ).
- Chain Rule — Rule for differentiating composites: Derivative of outer × derivative of inner.
- GCF (Greatest Common Factor) — The largest factor shared by all terms, used for factoring expressions.
- Rationalizing the Denominator — Process to eliminate square roots from the denominator.
Action Items / Next Steps
- Practice applying the product and chain rules to other polynomial and radical functions.
- Review how to factor out the greatest common factor from expressions.
- Complete any assigned homework problems involving derivatives and simplification.