Implicit differentiation

Implicit differentiation finds derivatives from equations relating variables without first solving explicitly for one variable as a function of another. It applies the chain rule to dependent variables.

Chain rule: A differentiation rule for composing functions: the derivative of a composition multiplies the derivatives of its successive functions. Differentiating a term involving the dependent variable requires multiplying by that variable’s derivative.
Function: A mapping that assigns each input in its domain exactly one output. An implicit equation can describe a function locally even when it does not display one explicitly.
Circle: The plane curve consisting of points at a fixed distance from a center. Its equation yields tangent slopes without solving separately for the upper and lower semicircles.
Explicit function: A function presented with its output directly expressed in terms of its input. Implicit differentiation avoids first rearranging a relation into this form.
Related rates: A method for finding how quickly quantities change together by differentiating an equation that relates them with respect to time. It uses the same equation-based differentiation when several quantities vary with time.
Equation: A mathematical statement that two expressions are equal. The method begins with an equation constraining the values of multiple variables.
Ellipse: A closed plane curve whose points have a constant sum of distances from two foci. Its defining equation can be differentiated directly to find tangent slopes.
Logarithmic differentiation: A technique that differentiates the logarithm of a function to simplify products, quotients, or variable exponents. It is another strategic way to differentiate expressions, but starts from an explicit function.
Derivative: A function’s instantaneous rate of change, defined as the limit of its average rate of change over shrinking intervals. Implicit differentiation’s goal is to determine a derivative from a relation.
Differentiability: The property of a function having a derivative at a point or throughout a specified domain. A smooth local curve is needed for its slope to be described by a derivative.

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