دروس في التفاضل والتكامل

Study Notes

Calculus: The Study of Change

Calculus is a branch of mathematics that deals with continuous change, often referred to as infinitesimal calculus. It was founded by Sir Isaac Newton and G.W. Leibniz in the 17th century. The discipline is primarily divided into two parts: differential calculus and integral calculus, each focusing on a different aspect of change.

Differential Calculus

Differential calculus is the subfield of calculus that deals with derivatives and differentials. It is concerned with how functions change as the input (often denoted as x) varies. The primary operation in differential calculus is the derivative, which gives the infinitesimal change of a function with respect to one of its variables. The derivative represents the slope of the function at every point it is defined.

Rules of Differentiation

Calculus employs several rules to find the derivative of a function. These rules include:

Constant times a function rule: If f(x) = c * g(x), where c is a constant and g(x) is a function, the derivative of f(x) is the product of the constant and the derivative of g(x).
Power rule: If f(x) = x^n, the derivative of f(x) is f'(x) = n * x^(n-1).
Product rule: If f(x) = g(x) * h(x), the derivative of f(x) is f'(x) = g'(x) * h(x) + g(x) * h'(x).
Quotient rule: If f(x) = g(x) / h(x), the derivative of f(x) is f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / h(x)^2.
Chain rule: If f(x) = g(h(x)), the derivative of f(x) is f'(x) = g'(h(x)) * h'(x).

These rules provide a systematic way to compute the derivative of a function, allowing us to analyze the function's behavior at any point. They are fundamental to understanding how functions change, and they have numerous applications in various fields.

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