Implicit Differentiation: Finding dy/dx in Harder Cases

Calculus 1 · Axiom Academy

LESSON Finding dy/dx in Harder Cases Beyond the circle — implicit differentiation when x and y are multiplied, wrapped in trig or exponentials, and when dy/dx hides on both sides. 1. When x and y Are Multiplied On x^2 + y^2 = 25 , every term is either an x or a y — never both. The moment they share a term, like in xy = 6 , you can't differentiate it as one block. You need the product rule , and the y -factor still demands the chain rule (so it drags a along). The product rule: derivative of the first times the second, plus the first times the derivative of the second 2. Trig Functions of Mixed Expressions An equation like looks intimidating, but it's just two chain rules stacked. The outer chain rule peels off the sine; the inner derivative — which is a product rule, because xy mixes both variables — handles what's inside, and that's where the rides along. 3. Exponentials and Logarithms Exponentials and logs are friendly — their derivatives are clean — but only if you remember the chain rule on every y . Writing is wrong when y depends on x . The correct version drops a : — same base, times the chain-rule tag. — derivative of the outer, evaluated at y , times the tag. 4. When dy/dx Appears Twice: Isolating It Here's the part that trips students up. In harder equations, shows up in more than one term after you differentiate. To solve for it you collect, factor, and divide — it's just algebra, but the steps must be deliberate. Watch x^2 y + x y^2 = 12 go end to end.

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