∫ f(x)nf'(x) dx = f(x)n + 1/(n + 1) + C
n ≠-1
C is the constant of integration.
∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g/(x) dx This is known as integration by parts.
d/dx ∫ f(x) dx = f(x)
∫ d/dx f(x) dx = f(x) + C C is the constant of integration.
∫ af(x) dx = a ∫ f(x) dx
∫ f'(x)/f(x) dx = ln(f(x)) + C C is the constant of integration.
The derivative refers to the rate at which a function changes with respect to another measure. The differential refers to the actual change in a function across a parameter. The differential of a function is equal to its derivative multiplied by the differential of the independent variable . The derivative of a function is the the LIMIT of the ratio of the increment of a function to the increment of the independent variable as the latter tends to zero.
∫ f'(x)(af(x) + b)n dx = (af(x) + b)n + 1/[a(n + 1)] + C C is the constant of integration.
∫ f'(x)/√(af(x) + b) dx = 2√(af(x) + b)/a + C C is the constant of integration.
You can differentiate a function when it only contains one changing variable, like f(x) = x2. It's derivative is f'(x) = 2x. If a function contains more than one variable, like f(x,y) = x2 + y2, you can't just "find the derivative" generically because that doesn't specify what variable to take the derivative with respect to. Instead, you might "take the derivative with respect to x (treating y as a constant)" and get fx(x,y) = 2x or "take the derivative with respect to y (treating x as a constant)" and get fy(x,y) = 2y. This is a partial derivative--when you take the derivative of a function with many variable with respect to one of the variables while treating the rest as constants.
If y is a differentiable function of u, and u is a differentiable function of x. Then y has a derivative with respect to x given by the formula : dy/dx = dy/du. du/dx This formula is known as the Chain Rule and says, " The rate of change of y with respect to x is the rate of change of y with respect to u multiplied by the rate of change of u with respect to x."
The derivative with respect to 'x' is 4y3 . The derivative with respect to 'y' is 12xy2 .
∫ f'(x)/√[f(x)2 + a] dx = ln[f(x) + √(f(x)2 + a)] + C C is the constant of integration.