Implicit & Inverse Differentiation Cheat Sheet

The core ideas of Implicit & Inverse Differentiation distilled into a single, scannable reference — perfect for review or quick lookup.

PiqCue — piqcue.com/implicit-inverse-differentiation/cheatsheet

Quick Reference

Chain Rule

The derivative of a composite function f(g(x)) equals f'(g(x)) times g'(x). It is the foundational technique for differentiating compositions, implicit equations, and inverse functions.

Implicit Differentiation

A technique for finding dy/dx when y is not isolated as an explicit function of x. Differentiate both sides of the equation with respect to x, applying the chain rule to y-terms (since y depends on x), then solve for dy/dx.

Derivative of Inverse Trigonometric Functions

Formulas for derivatives of arcsin, arccos, arctan, and other inverse trig functions, derived using implicit differentiation and the Pythagorean identity.

Derivative of Inverse Functions

If f and g are inverse functions, then g'(x) = 1/f'(g(x)). This formula lets you find the derivative of an inverse function at a point without finding the inverse explicitly.

Higher-Order Derivatives

The second derivative f''(x) is the derivative of f'(x) and measures concavity and rate of change of slope. In implicit differentiation, finding the second derivative often requires substituting the first derivative expression back in.

Logarithmic Differentiation

A technique for differentiating functions with variables in both the base and exponent, or products/quotients with many factors. Take the natural log of both sides, differentiate implicitly, then solve for dy/dx.

Key Terms at a Glance

Chain Rule:Rule for differentiating composites: d/dx[f(g(x))] = f_prime(g(x))*g_prime(x).

Implicit Differentiation:Differentiating when y is not solved explicitly, treating y as f(x).

Inverse Function Theorem:g_prime(y) = 1/f_prime(g(y)) when f_prime(g(y)) is nonzero.

Composite Function:f(g(x)) - one function applied to result of another.

Inverse Trig Functions:arcsin, arccos, arctan, arcsec, arccsc, arccot.

Logarithmic Differentiation:Take ln of both sides before differentiating. For variable exponents.

Higher-Order Derivatives:Derivatives of derivatives: second, third, etc.

Explicit Function:y written directly as f(x).

Implicit Equation:Equation relating x and y not solved for y.

Inner Function:In f(g(x)), g(x) is evaluated first.

Outer Function:In f(g(x)), f is applied to g(x).

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