The function is defined for all real numbers .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

The function g(x)=x2/3g(x)=x^{2/3} is defined for all real numbers xx. (a) Compute g′(x)g'(x) for x≠0x\ne 0. (b) Evaluate lim⁡x→0g′(x)\displaystyle \lim_{x\to 0} g'(x), or explain why the limit does not exist. (c) Determine whether the graph of gg possesses a tangent line at the origin. If so, give its equation; if not, describe the geometric behavior of the graph near the origin that prevents the existence of a tangent line. (d) Find the equation of the line normal to the graph of gg at the origin, if such a line exists, and justify your answer.

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