Let . Which of the following expressions equals…

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let h(x)=arctan⁡ ⁣(arcsin⁡x1−x2)h(x)=\arctan\!\left(\dfrac{\arcsin x}{\sqrt{1-x^{2}}}\right). Which of the following expressions equals h′(x)h'(x)? (a) 1+xarcsin⁡x1−x21−x2[ 1+(arcsin⁡x)21−x2]\displaystyle \frac{1+\dfrac{x\arcsin x}{\sqrt{1-x^{2}}}}{1-x^{2}}\left[\,1+\dfrac{(\arcsin x)^{2}}{1-x^{2}}\right] (b) 1+xarcsin⁡x1−x21−x2\displaystyle \frac{1+\dfrac{x\arcsin x}{\sqrt{1-x^{2}}}}{1-x^{2}} (c) 1+xarcsin⁡x1−x2(1−x2)[ 1+(arcsin⁡x)21−x2]\displaystyle \frac{1+\dfrac{x\arcsin x}{\sqrt{1-x^{2}}}}{(1-x^{2})\left[\,1+\dfrac{(\arcsin x)^{2}}{1-x^{2}}\right]} (d) 1−xarcsin⁡x1−x2(1−x2)[ 1+(arcsin⁡x)21−x2]\displaystyle \frac{1-\dfrac{x\arcsin x}{\sqrt{1-x^{2}}}}{(1-x^{2})\left[\,1+\dfrac{(\arcsin x)^{2}}{1-x^{2}}\right]}

Related theory

Browse the problem library