Derivatives — Rules, Chain Rule, Implicit Differentiation

GRE Math Subject · Axiom Academy

LESSON Derivatives — Rules, Chain Rule, Implicit Differentiation Master differentiation techniques essential for the exam Step 1: Core Differentiation Rules Basic derivatives (memorize these!): Step 2: Chain Rule (The Most Important) Chain Rule: If y = f(g(x)), then Strategy: Identify the "outside" and "inside" functions. Differentiate the outside, keep the inside, multiply by the derivative of the inside. Step 3: Implicit Differentiation When y is defined implicitly (e.g., x² + y² = 25), use implicit differentiation. 1. Differentiate both sides with respect to x 2. Remember that y is a function of x, so use chain rule on y terms: d/dx[y²] = 2y(dy/dx) 3. Collect dy/dx terms on one side Example: Find dy/dx from x² + y² = 25 Implicit differentiation is used to find slopes on curves where y isn't solved for x explicitly. Step 4: Higher Derivatives and Special Cases Logarithmic Differentiation (for messy products/quotients): Take ln of both sides, differentiate implicitly, solve for y' Why logarithmic differentiation helps: Instead of using product rule on f(x)·g(x)·h(x), take ln and get ln(f) + ln(g) + ln(h), which is easier to differentiate. Logarithmic method: y = x^x ⟹ ln(y) = x·ln(x)

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