(0:00) Given a demand curve (quantity sold as a function of price) q = f(p), the revenue is R = g(p) = p*f(p). The condition dR/dp = 0 is equivalent, by the Product Rule, to p*f'(p) + f(p) = 0, which means f'(p) = -f(p)/p = -(f(p)-0)/(p-0). The maximum revenue therefore will occur when the slope of the tangent line to f(p) equals the opposite of the slope of the line connecting (0,0) and (p,f(p)). (5:08) Show slide with the derivation and statement of the Quotient Rule. (8:11) Example: find the derivative of tan(x) with the Quotient Rule. (11:07) Example: differentiate the rational function f(x) = (4x + 3)/(5x - 2) with the Quotient Rule to see that it is always decreasing on its domain (its derivative is always negative), in spite of its vertical asymptote. (14:36) Show slide with the derivation and the statement of the Chain Rule (there's a mistake on the slide). (16:49) Example: find the derivative of h(x) = sin(x^2) with the Chain Rule. (18:57) Example: find the derivative of f(x) = sin^(2)(x) = (sin(x))^2 with the Chain Rule. (21:38) Example: find the derivative of f(x) = x*e^(x^(2)). Need the Product Rule first, then the Chain Rule. This function is always increasing (its derivative is always positive).
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(0:00) Given a demand curve (quantity sold as a function of price) q = f(p), the revenue is R = g(p) = p*f(p). The condition dR/dp = 0 is equivalent, by the Product Rule, to p*f'(p) + f(p) = 0, which means f'(p) = -f(p)/p = -(f(p)-0)/(p-0). The maximum revenue therefore will occur when the slope of the tangent line to f(p) equals the opposite of the slope of the line connecting (0,0) and (p,f(p)). (5:08) Show slide with the derivation and statement of the Quotient Rule. (8:11) Example: find the derivative of tan(x) with the Quotient Rule. (11:07) Example: differentiate the rational function f(x) = (4x + 3)/(5x - 2) with the Quotient Rule to see that it is always decreasing on its domain (its derivative is always negative), in spite of its vertical asymptote. (14:36) Show slide with the derivation and the statement of the Chain Rule (there's a mistake on the slide). (16:49) Example: find the derivative of h(x) = sin(x^2) with the Chain Rule. (18:57) Example: find the derivative of f(x) = sin^(2)(x) = (sin(x))^2 with the Chain Rule. (21:38) Example: find the derivative of f(x) = x*e^(x^(2)). Need the Product Rule first, then the Chain Rule. This function is always increasing (its derivative is always positive).
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