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:
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} ) Then checked the solution in the back: —
[ \frac{d}{dx}[x^2 y^3] + \frac{d}{dx}[\sin(y)] = \frac{d}{dx}[5x] ] Leo’s text back: “Why didn’t you give me this sooner
Derivative of (\sin(y)): ( \cos(y) \frac{dy}{dx} ) then ( y = u^3 )