No panic. No algebra mistake. Just solid, drilled-in calculus skills. Mia scored 86% on the final. Her overall grade rose to a B+. More importantly, she stopped fearing calculus — she started enjoying the precision.
Mia wasn’t amused. The problem wasn’t understanding big ideas — limits, derivatives, integrals made sense in lecture. It was the mechanics . Chain rule with nested exponentials? Implicit differentiation gone wrong? Definite integrals where she’d forget the constant? Little errors snowballed into wrong answers.
She later recommended McMullen’s workbook to Leo, who was struggling with integration. Leo’s text back: “Why didn’t you give me this sooner?” If you’d like to practice in McMullen’s direct style, here are three problems with full solutions: 1. Power Rule & Negative Exponents Problem : Differentiate ( f(x) = \frac{5}{x^3} - 2\sqrt{x} ) No panic
Right side: ( 5 )
Then checked the solution in the back: — ( y = [\sin(4x)]^3 ) Let ( u = \sin(4x) ), then ( y = u^3 ), ( \frac{dy}{du} = 3u^2 ) ( \frac{du}{dx} = \cos(4x) \cdot 4 ) (chain rule again inside) ( \frac{dy}{dx} = 3[\sin(4x)]^2 \cdot 4\cos(4x) = 12\sin^2(4x)\cos(4x) ) ✓ She had gotten it right — but the solution reminded her to explicitly show the inner chain rule on (4x), a step she often rushed. A Week Later — The Improvement Mia did two chapters per night. On Wednesday, she tackled implicit differentiation : Problem 47 — Find ( \frac{dy}{dx} ) for ( x^2 y^3 + \sin(y) = 5x ) She wrote: Mia scored 86% on the final
So: ( 2x y^3 + 3x^2 y^2 \frac{dy}{dx} + \cos(y) \frac{dy}{dx} = 5 )
Group (\frac{dy}{dx}) terms: ( \frac{dy}{dx} (3x^2 y^2 + \cos y) = 5 - 2x y^3 ) Mia wasn’t amused
Derivative of (\sin(y)): ( \cos(y) \frac{dy}{dx} )