Overview
This lecture introduces two important differentiation rules—the product rule and the quotient rule—expanding the set of tools for finding derivatives of complex functions.
Differentiation Toolbox Review
- The definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} )
- The derivative of a constant is zero.
- The power rule: ( (x^n)' = n x^{n-1} ), where ( n ) can be any real number.
- The constant multiple rule: ( (cf)' = c f' ) for a constant ( c ).
- Sum/difference rule: ( (f \pm g)' = f' \pm g' ).
- Chain rule: ( (f(g(x)))' = f'(g(x)) \cdot g'(x) ).
Introducing the Product Rule
- The product rule: ( (fg)' = f g' + g f' ).
- When differentiating a product, multiply the first function by the derivative of the second and add the second times the derivative of the first.
Product Rule Examples
- For ( f(x) = x^2 + 3x - 1 ) and ( g(x) = 2x^2 - x - 5 ): Apply the product rule and simplify the result.
- When one factor involves a composite function (e.g., a square root), use the chain rule within the product rule.
Introducing the Quotient Rule
- The quotient rule: ( \left(\frac{f}{g}\right)' = \frac{g f' - f g'}{g^2} ).
- When differentiating a quotient, subtract the product of the numerator and derivative of the denominator from the product of the denominator and derivative of the numerator, then divide by the denominator squared.
Quotient Rule Examples
- For ( y = \frac{2x-3}{4x+1} ), apply the quotient rule and simplify.
- For composite functions or powers in numerator/denominator, use the chain rule as needed.
- In complex cases, the product and chain rules may be nested within the quotient rule computation.
Key Terms & Definitions
- Product Rule — The derivative of ( f(x)g(x) ) is ( f(x)g'(x) + g(x)f'(x) ).
- Quotient Rule — The derivative of ( f(x)/g(x) ) is (f’(x)-f(x)g’(x))/g(x)²
- Chain Rule — Used when differentiating a composite function: ( (f(g(x)))' = f'(g(x))(g’(x))
- Point of Inflection — Where the concavity of a function changes, determined by the sign change in the second derivative.
Action Items / Next Steps
- Practice product and quotient rule problems, especially with functions involving chain rule.
- For homework, identify and analyze points of inflection for assigned functions.
- Prepare for further study on how derivatives reveal function behavior.