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The chain rule for differentiation applies when you have a composite function. Multiply the derivative of the outer function by the derivative of the inner function. This technique is essential for derivatives involving nested functions, exponentials, and trigonometric compositions.

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations. It connects rates of change to accumulated quantities, enabling you to evaluate definite integrals using antiderivatives and solve real-world problems involving area, volume, and motion.

Integration techniques like substitution and integration by parts let you transform complex integrals into simpler forms. Substitution reverses the chain rule; integration by parts handles products of functions. Both are practical for calculating areas, volumes, and solving differential equations.

Partial derivatives measure how a multivariable function changes with respect to one variable. The gradient vector combines all partial derivatives, pointing in the direction of steepest ascent. These concepts are critical for optimization, physics applications, and understanding surfaces in three dimensions.

Taylor and Maclaurin expansions represent functions as infinite polynomial series centered at a point. These approximations enable you to evaluate complex functions numerically, solve differential equations, and analyze function behavior near specific points with controlled precision.

Yes. Calculus optimization uses derivatives to find maximum and minimum values of functions. By setting derivatives to zero and analyzing critical points, you can solve practical problems in economics, engineering, and physics involving constrained and unconstrained optimization.