Find the derivative using the first principle f(x)=3x^2−x+4 - Derivatives - Differentiation #48

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Differentiation is a mathematical process that finds the rate of change (also known as the derivative) of a function at a specific point. It's a fundamental concept in calculus that allows for the study of motion, optimization, and analysis of functions. The derivative can be used to identify the maximum and minimum points on a curve, and to determine the slope of the curve at a particular point. Differentiation also plays a crucial role in solving many real-world problems, such as finding the minimum time required to reach a certain distance.

Rules of differentiation

The rules of differentiation are mathematical formulas used to find the derivative of a function. Here are some common rules:

Power rule: For a function f(x) = x^n, the derivative is given by d/dx (x^n) = nx^(n-1).

Sum rule: For two functions f(x) and g(x), the derivative of their sum is given by d/dx (f(x) + g(x)) = d/dx (f(x)) + d/dx (g(x)).

Product rule: For two functions f(x) and g(x), the derivative of their product is given by d/dx (f(x)g(x)) = f(x) d/dx (g(x)) + g(x) d/dx (f(x)).

Quotient rule: For two functions f(x) and g(x), the derivative of their quotient is given by d/dx (f(x)/g(x)) = (g(x) d/dx (f(x)) - f(x) d/dx (g(x))) / g^2(x).

Chain rule: If y = g(u) and u = f(x), then d/dx (y) = d/du (g(u)) * d/dx (u).

These are just a few examples of the rules of differentiation. There are many more rules and techniques used in finding derivatives.

The chain rule is a rule in calculus that allows one to find the derivative of a composite function. Given two functions, f(x) and g(x), the composite function h(x) = f(g(x)) is formed by first applying g(x) to x, and then applying f(x) to the result. The chain rule states that the derivative of h(x) with respect to x is given by the derivative of f(g(x)) with respect to g(x) multiplied by the derivative of g(x) with respect to x, or mathematically:

dh/dx = df/dg * dg/dx.

In other words, the chain rule says that to find the derivative of a composite function, we must first find the derivative of the inner function and then multiply that result by the derivative of the outer function. The chain rule is commonly used in finding derivatives of functions involving exponentials, logarithms, and inverse functions.