Implicit Differentiation

Calculus 3 · Axiom Academy

LESSON Implicit Differentiation Differentiating relations like F(x,y,z)=c with the chain rule — and reading the answer straight off the gradient . Given a relation F(x,y)=0 , we want without solving for y . The key move: treat y as a hidden function of x and differentiate every term with the chain rule. Staying on the curve forces y to respond to each change in x . Worked case — the circle x^2+y^2=25 Differentiate both sides in x : . The term is the chain rule at work. For a general relation F(x,y)=0 there is one clean formula, built from the two partial derivatives of F — the components of the gradient . Why it works. Along the curve, F never changes, so its total differential is zero: Dividing by dx and rearranging gives — the horizontal contribution and the vertical contribution must cancel. The gradient is always perpendicular to the level curve F(x,y)=c , and it points toward increasing F (outward, across the family of level curves). That single fact is the formula. Perpendicular to the curve; points in the direction of greatest increase of F . Has direction , running along the curve at each point. 4. Extension to Three Variables For a surface F(x,y,z)=0 , the variable z is defined implicitly as a function of x and y . The same principle gives both partial derivatives — and the implicit-function theorem guarantees we can solve for z locally wherever . Worked case — the sphere x^2+y^2+z^2=9 Let F(x,y,z)=x^2+y^2+z^2-9 , so . Then the two partials of z are:

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