c’ = 0, where c is a constant

(x^n)’ = nx^(n-1), where n is a real number and n ≠ 0.

(a^x)’ = a^x * ln(a), where a is a positive constant.

(log_a(x))’ = 1 / (x * ln(a)), where a is a positive constant

(sin(x))’ = cos(x)

(cos(x))’ = -sin(x)

(tan(x))’ = sec^2(x) = 1/cos^2(x)

(arcsin(x))’ = 1 / √(1 - x^2), where |x| < 1

(arccos(x))’ = -1 / √(1 - x^2), where |x| < 1

(arctan(x))’ = 1 / (1 x^2)

Addition rule (sum rule): If there are two differentiable functions f(x) and g(x), then the derivative of their sum (or difference) is equal to the sum (or difference) of their respective derivatives: (f(x) ) ± g(x))' = f'(x) ± g'(x)

Multiplication rule (product rule): The derivative of the product of two differentiable functions f(x) and g(x) is equal to the derivative of the first function times the second function, plus the second function times the first Derivative of a function: (f(x)g(x))' = f'(x)g(x) f(x)g'(x)

Division rule (quotient rule): The derivative of the quotient of two differentiable functions f(x) and g(x) is equal to the derivative of the numerator times the denominator minus the derivative of the numerator times the denominator, and then divided by the square of the denominator: (f (x)/g(x))' = (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2

Chain rule (composite function rule): If there is a composite function h(x) = f(g(x)), where f(x) and g(x) are both differentiable, then the derivative of this composite function is equal to The derivative of the outer function with respect to the inner function is multiplied by the derivative of the inner function: h'(x) = f'(g(x))g'(x)

Boundedness: If there is a constant M > 0 such that |f(x)| ≤ M for all x belonging to the domain, the function is said to be bounded.

Monotonicity: If for any two numbers x1 and x2 in the domain, when x1 < x2, f(x1) ≤ f(x2), the function is said to be monotonically increasing; if both f(x1) ≥ f( x2), then the function is said to be monotonically decreasing.

Parity: If f(-x) = -f(x) exists for any x in the domain of definition, the function is called an odd function; if f(-x) = f(x) exists, the function is called an odd function. even function.

Periodicity: If there is a non-zero constant T such that f(x T) = f(x) for any x in the domain of definition, then the function is said to be periodic, and T is called the period of the function.

Sine function: f(x) = sin(x)

Cosine function: f(x) = cos(x)

Tangent function: f(x) = tan(x)

Cotangent function: f(x) = cot(x)

Secant function: f(x) = sec(x)

Cosecant function: f(x) = csc(x)

Arcsine function: f(x) = arcsin(x) or sin^(-1)(x)

Arccosine function: f(x) = arccos(x) or cos^(-1)(x)

Arctangent function: f(x) = arctan(x) or tan^(-1)(x)

Inverse cotangent function: f(x) = arccot(x) or cot^(-1)(x)

Arcsec function: f(x) = arcsec(x) or sec^(-1)(x)

Inverse cosecant function: f(x) = arccsc(x) or csc^(-1)(x)