AP

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let y=ln⁡((1+x2)3/2)y = \ln\bigl((1 + x^2)^{3/2}\bigr). (i) Show that dydx=3x1+x2\dfrac{dy}{dx} = \dfrac{3x}{1 + x^2}. (ii) Prove that d2ydx2=3(1−x2)(1+x2)2\dfrac{d^2y}{dx^2} = \dfrac{3(1 - x^2)}{(1 + x^2)^2}. (iii) Hence state the intervals on which the curve is concave upward and concave downward.

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