Calculus: Derivatives and Differentiation

The derivative of a function f(x) is the instantaneous rate of change of f(x) with respect to x. It represents the slope of the tangent line to the graph of f(x) at a given point.

The process of finding the derivative of a function is called differentiation.

The derivative of a function f(x) at a point x = c is the limit as h approaches 0 of the difference quotient (f(c + h) - f(c))/h, provided this limit exists.

A function is differentiable at a point if its derivative exists at that point.

A function is differentiable on an interval if it is differentiable at every point in the interval.

The derivative of a function f(x) is denoted by various notations, including f'(x), df/dx, and Dy.

A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the value of the function at that point.

Theorem I states that if a function is differentiable at a point, then it is continuous at that point.

The converse of Theorem I is false. A function can be continuous at a point but not differentiable at that point.

The derivative of a constant function is always zero. For example, the derivative of y = 5 is y' = 0.

The derivative of a linear function is its slope. For example, the derivative of y = 2x + 3 is y' = 2.

The power rule states that the derivative of x^n is nx^(n-1) where n is any real number.

The sum and difference rule states that the derivative of a sum or difference of functions is the sum or difference of their derivatives, respectively.

The product rule states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

The quotient rule states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

The second derivative of a function f(x) is the derivative of its first derivative. It is denoted by f''(x), d^2y/dx^2, or D^2y.

The third derivative of a function f(x) is the derivative of its second derivative. It is denoted by f'''(x), d^3y/dx^3, or D^3y.

Higher order derivatives are derivatives of higher order than the second or third derivative. They are denoted by f^(n)(x), d^ny/dx^n, or D^ny.

The second derivative can be interpreted as the rate of change of the slope of the tangent line to the graph of f(x) at each point.

The second derivative can reveal whether the graph of a function bends upward or downward from the tangent line as we move off the point of tangency.

The third derivative can provide information about the rate of change of the concavity of the graph of a function.

The tangent line to the graph of a function at a point is the line that best approximates the function near that point.

The slope of the tangent line at a point represents the instantaneous rate of change of the function at that point.

The derivative of a function can be used to find the equation of the tangent line to the graph of the function at a given point.

The derivative can be used to analyze the behavior of a function, such as finding its critical points, intervals of increase and decrease, and points of inflection.

The derivative can be used in optimization problems to find the maximum or minimum values of a function.

The derivative can be used in related rates problems to find the rate of change of one variable with respect to another.

The derivative can be used in approximation problems to estimate the value of a function near a known point.