Differentiation and Integration Formulas
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These flashcards cover key differentiation and integration formulas to aid in understanding calculus.
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40 Terms
Differentiation Formula for a constant
The derivative of a constant k is 0.
Sum Rule of Differentiation
The derivative of the sum of two functions f(x) and g(x) is the sum of their derivatives.
Product Rule of Differentiation
The derivative of the product of two functions f(x) and g(x) is given by f(x)g'(x) + g(x)f'(x).
Quotient Rule of Differentiation
The derivative of the quotient of two functions f(x) and g(x) is given by (g(x)f'(x) - f(x)g'(x)) / (g(x))^2.
Chain Rule of Differentiation
The derivative of a composite function f(g(x)) is f'(g(x))g'(x).
Power Rule of Differentiation
The derivative of x raised to the power n is nx^(n-1).
Derivative of sin x
The derivative of sin x is cos x.
Derivative of cos x
The derivative of cos x is -sin x.
Derivative of tan x
The derivative of tan x is sec^2 x.
Derivative of cot x
The derivative of cot x is -csc^2 x.
Derivative of sec x
The derivative of sec x is sec x tan x.
Derivative of csc x
The derivative of csc x is -csc x cot x.
Derivative of e^x
The derivative of e^x is e^x.
Derivative of a^x
The derivative of a^x is a^x ln a.
Derivative of ln |x|
The derivative of ln |x| is 1/x.
Derivative of sin^(-1)x
The derivative of sin^(-1)x is 1/√(1 - x^2).
Derivative of cos^(-1)x
The derivative of cos^(-1)x is -1/√(1 - x^2).
Derivative of tan^(-1)x
The derivative of tan^(-1)x is 1/(x^2 + 1).
Derivative of cot^(-1)x
The derivative of cot^(-1)x is -1/(x^2 + 1).
Derivative of sec^(-1)x
The derivative of sec^(-1)x is 1/|x|√(x^2 - 1).
Derivative of csc^(-1)x
The derivative of csc^(-1)x is -1/|x|√(x^2 - 1).
Integration of dx
The integral of dx is x + C.
Integration of x^n
The integral of x^n is x^(n+1)/(n + 1) + C.
Integration of 1/x
The integral of 1/x is ln |x| + C.
Integration of e^x
The integral of e^x is e^x + C.
Integration of a^x
The integral of a^x is 1/(ln a) a^x + C.
Integration of ln x
The integral of ln x is x ln x - x + C.
Integration of sin x
The integral of sin x is -cos x + C.
Integration of cos x
The integral of cos x is sin x + C.
Integration of tan x
The integral of tan x is -ln |cos x| + C.
Integration of cot x
The integral of cot x is ln |sin x| + C.
Integration of sec x
The integral of sec x is ln |sec x + tan x| + C.
Integration of csc x
The integral of csc x is -ln |csc x + cot x| + C.
Integration of sec^2 x
The integral of sec^2 x is tan x + C.
Integration of csc^2 x
The integral of csc^2 x is -cot x + C.
Integration of sec x tan x
The integral of sec x tan x is sec x + C.
Integration of csc x cot x
The integral of csc x cot x is -csc x + C.
Integration involving √(a^2 - x^2)
The integral of 1/√(a^2 - x^2) is sin^(-1)(x/a) + C.
Integration involving (a^2 + x^2)
The integral of 1/(a^2 + x^2) is (1/a) tan^(-1)(x/a) + C.
Integration involving x/√(x^2 - a^2)
The integral of x/√(x^2 - a^2) is (1/a) sec^(-1)(|x|/a) + C.