Differentiating When y Isn’t Isolated: Implicit and Higher-Order Derivatives

Implicit relationship

An equation connecting x and y where y is not isolated (e.g., x^2 + y^2 = 25), describing a curve even if it is not a function y = f(x) globally.

Implicit differentiation

A technique for finding dy/dx by differentiating both sides of an equation with respect to x without first solving for y.

Treat y as a function of x

The implicit-differentiation mindset that y depends on x along the curve, so derivatives of expressions involving y require the Chain Rule.

Chain Rule “receipt” factor

When differentiating an expression involving y with respect to x, you multiply by dy/dx to account for y changing as x changes.

Derivative of y^2 (implicit)

d/dx(y^2) = 2y·(dy/dx).

Derivative of y^3 (implicit)

d/dx(y^3) = 3y^2·(dy/dx).

Derivative of sin(y) (implicit)

d/dx(sin(y)) = cos(y)·(dy/dx).

Derivative of cos(y) (implicit)

d/dx(cos(y)) = −sin(y)·(dy/dx).

dy/dx notation

The derivative of y with respect to x; in implicit problems it represents the slope of the tangent line in terms of x and y.

y′ notation

An alternative notation for dy/dx; it means the same first derivative.

d/dx[F] notation

Operator notation meaning “differentiate F with respect to x,” useful when differentiating both sides of an equation.

Implicit differentiation algorithm

Differentiate both sides w.r.t. x, apply Chain Rule to y-terms, collect all dy/dx terms on one side, factor out dy/dx, and solve.

Product Rule for xy (implicit setting)

d/dx(xy) = x·(dy/dx) + y, because y depends on x.

Circle slope via implicit differentiation

For x^2 + y^2 = 25, implicit differentiation gives dy/dx = −x/y.

Tangent line (point-slope form)

A line using slope m and point (x1,y1): y − y1 = m(x − x1), often with m found from dy/dx.

Tangent line at (3,4) on x^2 + y^2 = 25

Since dy/dx = −x/y, the slope at (3,4) is −3/4, so y − 4 = (−3/4)(x − 3).

Example: implicit derivative of xy + sin(y) = x^2

Differentiating gives x(dy/dx) + y + cos(y)(dy/dx) = 2x, so dy/dx = (2x − y)/(x + cos(y)).

Common Chain Rule mistake (sin(y))

Incorrect: d/dx(sin(y)) = cos(y). Correct: cos(y)·(dy/dx).

Horizontal tangent condition (implicit)

If dy/dx = N(x,y)/D(x,y), a horizontal tangent occurs when N(x,y) = 0 and D(x,y) ≠ 0, at a point on the original curve.

Vertical tangent condition (implicit)

If dy/dx = N(x,y)/D(x,y), a vertical tangent occurs when D(x,y) = 0 and N(x,y) ≠ 0, at a point on the original curve.

Verification step for tangent candidates

After setting numerator or denominator conditions for horizontal/vertical tangents, you must check the candidate points satisfy the original equation.

Higher-order derivative (implicit context)

Any derivative beyond the first (e.g., y'' = d^2y/dx^2); found by differentiating y′ while remembering y and y′ depend on x.

Concavity via second derivative

If y'' > 0 the curve is concave up (slopes increasing); if y'' < 0 the curve is concave down (slopes decreasing).

Second derivative of the circle x^2 + y^2 = 25

Starting from y′ = −x/y, one form is y'' = −(y^2 + x^2)/y^3, which simplifies using x^2 + y^2 = 25 to y'' = −25/y^3.

Product Rule for x·y′ when finding y''

d/dx(x·(dy/dx)) = x·(d^2y/dx^2) + dy/dx, because both x and y′ depend on x.