Implicit Differentiation

Calculus 1 · Axiom Academy

LESSON Implicit Differentiation When you can't solve for y , differentiate the whole equation anyway — and let the chain rule do the work. 1. The Key Move: a y -Term Carries Differentiating an x -term is ordinary: . But y is itself a function of x , so y^2 is really the composite . The chain rule says: differentiate the outer power, then multiply by the derivative of the inside — and the inside's derivative is . A y -term: the chain rule attaches 2. Differentiate, Collect, Solve: the Circle Now the full procedure on x^2 + y^2 = 25 . Take of every term — the x^2 the normal way, the y^2 with the chain rule, the constant 25 to 0 . Then collect the terms and solve. The answer depends on both x and y , so it gives the slope at any point on the curve. Apply derivatives (chain rule on y^2 ): At (3, 4) the slope is ; at (0, 5) it is 0 — a horizontal tangent at the top of the circle. 3. When a Term Mixes x and y : the Folium The folium of Descartes x^3 + y^3 = 6xy can't be solved for y at all — implicit differentiation is the only way. One term, 6xy , mixes x and y , so it needs the product rule : (the second piece carries because y is a function of x ). Differentiate every term, gather the terms on one side, and factor. At the point (3, 3) on the loop, the slope is -1 — the curve is heading down at there. You can now find for a curve without ever solving it for y . Scroll up to revisit any step.

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