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Q: What is the integral of the derivative with respect to x of the function f multiplied by the quantity a times f plus b raised to the power of n with respect to x?

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

âˆ« [1/[f(x)(f(x) Â± g(x))]] dx = Â±âˆ«1/[f(x)g(x)] dx Â± (-1)âˆ« [1/[g(x)(f(x) Â± g(x))]] dx

The spacial derivative is the measure of a quantity as and how it is being changed in space. This is different from a temporal derivative and partial derivative.

Scalars are quantities that are described by a magnitude alone. A scalar quantity multiplied by a unit vector is not a scalar quantity but a vector quantity.

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

The spacial derivative is the measure of a quantity as and how it is being changed in space. This is different from a temporal derivative and partial derivative.

Increase in cost: take the first derivative with respect to the unit produced of a cost function. Total cost: sub-in the new quantity into the cost function.

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

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The cost of increasing the production by one unit. Mathematically, this can be derived as the derivative of the total costs with respect to quantity i.e. dc(q)/dq, where c(q) is the cost function and q is quantity.

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

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

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