Differential Calculus

Differential Calculus

• Successive Differentiation: a process of finding the derivative of a function multiple times, used to find higher-order derivatives.

Theorems in Differential Calculus

• Leibniz's Theorem: a formula for the nth derivative of a product of two functions, stated as: (uv)^n = Σ (nCk) * u^(n-k) * v^k.
• Rolle's Theorem: a fundamental theorem in calculus that states: if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that f'(c) = 0.
• Mean Value Theorem: a theorem that states: if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Expansions of Functions

• Taylor's Theorem: a theorem that states: any function f(x) can be represented as an infinite series, known as the Taylor series, around a point a, stated as: f(x) = f(a) + f'(a)(x-a) / 1! + f''(a)(x-a)^2 / 2! + ....
• Maclaurin's Theorem: a special case of Taylor's theorem, where the expansion is around the point a = 0, stated as: f(x) = f(0) + f'(0)x / 1! + f''(0)x^2 / 2! + ....

Indeterminate Forms

• L'Hospital's Rule: a rule used to evaluate the limit of an indeterminate form, such as 0/0 or ∞/∞, by finding the limit of the ratio of the derivatives of the numerator and denominator.

Differential Calculus

• Successive Differentiation is a method of finding higher-order derivatives of a function by repeatedly differentiating the function.
• Leibniz's Theorem states that the nth derivative of a product of two functions can be calculated using the formula: (uv)^n = u^n v + n u^(n-1) v' + ... + u v^n
• Rolle's Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = 0
• Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a)
• Taylor's Theorem states that any function can be represented as an infinite sum of terms, each term being a power of the variable of the function, with the first term being the function's value at a point, and each subsequent term being a successively higher derivative of the function at that point
• Maclaurin's Theorem is a special case of Taylor's Theorem, where the point of representation is 0, and the theorem states that any function can be represented as an infinite sum of terms, each term being a power of the variable of the function
• Indeterminate Forms can be evaluated using L'Hopital's Rule, which states that if the limit of a function as x approaches a certain value is of the form 0/0 or ∞/∞, then the limit can be evaluated by taking the derivative of the numerator and denominator separately and then finding the limit of the resulting expression.

Differential Calculus

• Successive Differentiation is a method of finding higher-order derivatives of a function by repeatedly differentiating the function.
• Leibniz's Theorem states that the nth derivative of a product of two functions can be calculated using the formula: (uv)^n = u^n v + n u^(n-1) v' + ... + u v^n
• Rolle's Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = 0
• Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a)
• Taylor's Theorem states that any function can be represented as an infinite sum of terms, each term being a power of the variable of the function, with the first term being the function's value at a point, and each subsequent term being a successively higher derivative of the function at that point
• Maclaurin's Theorem is a special case of Taylor's Theorem, where the point of representation is 0, and the theorem states that any function can be represented as an infinite sum of terms, each term being a power of the variable of the function
• Indeterminate Forms can be evaluated using L'Hopital's Rule, which states that if the limit of a function as x approaches a certain value is of the form 0/0 or ∞/∞, then the limit can be evaluated by taking the derivative of the numerator and denominator separately and then finding the limit of the resulting expression.