Q: What is the integral of the derivative with respect to x of a function of x divided by that same function of x with respect to x?

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d/dx âˆ« f(x) dx = f(x)

âˆ« f'(x)/âˆš(af(x) + b) dx = 2âˆš(af(x) + b)/a + C C is the constant of integration.

âˆ« f'(x)/âˆš[f(x)2 + a] dx = ln[f(x) + âˆš(f(x)2 + a)] + C C is the constant of integration.

âˆ« f'(x)/(p2 + q2f(x)2) dx = [1/(pq)]arctan(qf(x)/p)

âˆ« f'(x)/( q2f(x)2 - p2) dx = [1/(2pq)ln[(qf(x) - p)/(qf(x) + p)]

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d/dx âˆ« f(x) dx = f(x)

âˆ« d/dx f(x) dx = f(x) + C C is the constant of integration.

âˆ« f'(x)/âˆš(af(x) + b) dx = 2âˆš(af(x) + b)/a + C C is the constant of integration.

âˆ« f'(x)/âˆš[f(x)2 + a] dx = ln[f(x) + âˆš(f(x)2 + a)] + C C is the constant of integration.

Marginal cost - the derivative of the cost function with respect to quantity. Average cost - the cost function divided by quantity (q).

âˆ« f'(x)/(p2 + q2f(x)2) dx = [1/(pq)]arctan(qf(x)/p)

âˆ« f'(x)g(x) dx = f(x)g(x) - âˆ« f(x)g/(x) dx This is known as integration by parts.

âˆ« [f'(x)g(x) - g'(x)f(x)]/g(x)2 dx = f(x)/g(x) + C C is the constant of integration.

âˆ« f'(x)/( q2f(x)2 - p2) dx = [1/(2pq)ln[(qf(x) - p)/(qf(x) + p)]

âˆ« f(x)nf'(x) dx = f(x)n + 1/(n + 1) + C n â‰ -1 C is the constant of integration.

âˆ« [f'(x)g(x) - g'(x)f(x)]/[f(x)g(x)] dx = ln(f(x)/g(x)) + 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.