Derivatives — AI Study Guide

Mastering Derivatives

The derivative of f at x is the instantaneous rate of change of f with respect to x, defined as f'(x) = lim_{h→0} [f(x+h) - f(x)]/h. Geometrically, f'(x) is the slope of the tangent line to the graph of f at the point (x, f(x)). The derivative exists when this limit exists — at smooth points of the function. Points of non-differentiability include cusps, corners, and vertical tangents.

Differentiation rules allow computation of derivatives without using the limit definition every time. Fundamental rules: power rule (d/dx x^n = nx^{n-1}), product rule, quotient rule, and chain rule (for composite functions: d/dx f(g(x)) = f'(g(x))·g'(x)). Standard derivatives to memorize: d/dx sin x = cos x, d/dx cos x = -sin x, d/dx e^x = e^x, d/dx ln x = 1/x. Implicit differentiation handles equations where y cannot be solved explicitly.

Applications of derivatives are among the most important content in calculus. Related rates problems ask how rates of change of related quantities are connected. Optimization asks for the maximum or minimum of a function on an interval: find critical points (f'(x) = 0 or f'(x) undefined), check endpoints if the interval is closed, and use the first or second derivative test to classify critical points. The Mean Value Theorem connects average and instantaneous rates of change.

Higher-order derivatives represent rates of change of rates of change. The second derivative f''(x) measures the rate of change of slope. f''(x) > 0 means f is concave up (curve is cup-shaped); f''(x) < 0 means concave down (hill-shaped). Inflection points occur where concavity changes. In physics, if position is s(t), then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s''(t).

Frequently Asked Questions: Derivatives

What is the chain rule?

The chain rule gives the derivative of a composite function f(g(x)): d/dx[f(g(x))] = f'(g(x)) · g'(x). In words: derivative of the outside function (evaluated at the inside function), times the derivative of the inside function. Example: d/dx[sin(x²)] = cos(x²) · 2x. The chain rule is essential whenever one function is nested inside another — it applies repeatedly for more deeply nested compositions.

How do I find the maximum or minimum of a function?

For a function f on a closed interval [a,b]: (1) Find all critical points by setting f'(x) = 0 and solving, and identifying where f'(x) is undefined. (2) Evaluate f at all critical points in [a,b] and at the endpoints a and b. (3) The largest value is the absolute maximum; the smallest is the absolute minimum. For open intervals or all real numbers, critical points are still found by f'(x) = 0, and the second derivative test (f''(c) > 0 = local min; f''(c) < 0 = local max) classifies them.

How does Clario help with derivatives?

Clario processes your derivatives notes to generate flashcards covering differentiation rules, standard derivatives, and optimization techniques, an AI summary of derivative concepts and applications, and practice problems from your specific calculus course material.

Study Calculus with Clario

Upload your Calculus course notes and Clario generates a complete study system — summaries, flashcards, practice quizzes, and exam prep — in under 60 seconds. Every study tool is built from your specific notes, not generic content.

Try Clario Free

No credit card required. 3 free study packs.