Differentiation & Integration Formulas With Examples
Integration and differentiation formulas are essential topics in calculus, which explores how things change. Calculus is widely used in various fields, including science, economics, finance, and engineering. Differentiation involves calculating derivatives, which represent the slope or gradient of a curve (graph) at any specific point. The gradient of a curve at a given point is the slope of the tangent line drawn to that curve at that same point.
Understanding Integration
Integration refers to the process of calculating either a definite integral or an indefinite integral. For a real function f(x) defined over an interval [a, b] on the real line, the definite integral, represented as a∫b f(x), measures the area between the graph of the function, the horizontal axis, and the two vertical lines at the endpoints of that interval. When no specific interval is provided, it is termed an indefinite integral.
Differentiation Formulas
- Power Rule: (d/dx) (xn) = nxn – 1
- Derivative of a constant, a: (d/dx) (a) = 0.
- Derivative of a constant multiplied by a function f: (d/dx) (a · f) = a · f’
- Sum Rule: (d/dx) (f ± g) = f’ ± g’
- Product Rule: (d/dx) (fg) = fg’ + gf’
- Quotient Rule: (d/dx)(f/g) = (gf′ – fg′)/g².
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