Explain detailed differentiation and integration calculus with every key point covered till the level of JEE Advanced.

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Calculus is the mathematics of change, consisting of two fundamental branches. Differentiation finds instantaneous rates of change, like the slope of a tangent line at any point on a curve. Integration finds accumulated change, such as the area under a curve. These two operations are inverse processes, beautifully connected by the Fundamental Theorem of Calculus. The derivative measures the instantaneous rate of change of a function at any point. It's defined as the limit of the difference quotient as h approaches zero. Geometrically, this represents the slope of the tangent line to the curve. Watch as the secant line approaches the tangent line when h gets smaller. Key differentiation rules include the power rule and chain rule, which are essential for finding derivatives of complex functions. Integration is the reverse process of differentiation. The indefinite integral represents a family of antiderivatives, while the definite integral gives the exact area under a curve between two points. The Fundamental Theorem of Calculus beautifully connects differentiation and integration. Watch as Riemann rectangles approximate the area under the curve, and as we increase the number of rectangles, we approach the true integral value. Calculus has powerful applications across mathematics, physics, and engineering. Differentiation helps find maximum and minimum values for optimization problems, analyze related rates of change, and solve motion problems. Integration calculates areas between curves, volumes of complex solids, and work done by variable forces. These techniques are essential for JEE Advanced, where problems often combine multiple concepts requiring deep understanding of both differentiation and integration principles. To summarize what we've learned about calculus: Differentiation finds instantaneous rates of change and represents slopes of tangent lines. Integration calculates accumulated change and areas under curves. The Fundamental Theorem of Calculus beautifully connects these two inverse operations. Mastering the basic differentiation and integration rules is essential for JEE Advanced success, along with their applications in optimization and physics problems.