∫ f'(x)/√[f(x)2 + a] dx = ln[f(x) + √(f(x)2 + a)] + 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.
Constant is a quantity that does not change.
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.
∫ f'(x)/(p2 + q2f(x)2) dx = [1/(pq)]arctan(qf(x)/p)
∫ f'(x)/( q2f(x)2 - p2) dx = [1/(2pq)ln[(qf(x) - p)/(qf(x) + p)]
∫ f'(x)/√(af(x) + b) dx = 2√(af(x) + b)/a + C C is the constant of integration.
∫ f'(x)(af(x) + b)n dx = (af(x) + b)n + 1/[a(n + 1)] + 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).
∫ [f'(x)g(x) - g'(x)f(x)]/g(x)2 dx = f(x)/g(x) + C C is the constant of integration.
∫ [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.
∫ f'(x)/[f(x)√(f(x)2 - a2)] dx = (1/a)arcses(f(x)/a) + C C is the constant of integration.
Derivative of a constantThe derivative of any constant is zero. This can be easily conceptualized if you think of the graph of any constant value. The derivative can be thought of as the slope of the line tangent to a curve at any given point. If you graph the expression y = 3, for example, it is just a horizontal line intercepting the y axis at 3. The slope of that line is, of course, equal to zero, for any point on the curve (which in this case is a straight line). Therefore, the derivative (with respect to x) of y = 3 is zero. Since the slope of any horizontal line is zero, the derivative of any line of the form y = k, where k is a constant, is zero.Answer2:Any constant quantity and an expression that has a maximum or minimum or both, has a derivative equal to zero.
True
False
∫ f'(x)/√(a2 - f(x)2) dx = arcsin(f(x)/a) + C C is the constant of integration.
Constant is a quantity that does not change.
True!