Chapter 3 - Applications of Differentiation - 3.9 Exercises - Page 236: 18

$dy=-x^{2}(1-x^{2})^\frac{-1}{2}+(1-x^{2})^{\frac{1}{2}}\times dx$

$y=x\sqrt (1-x^{2})=x(1-x^{2})^{\frac{1}{2}}$ $\frac{dy}{dx}=(x)((\frac{1}{2})(1-x^{2})^\frac{-1}{2}(-2x))+(1)x(1-x^{2})^{\frac{1}{2}}$ (product rule, power rule, chain rule) $=-x^{2}(1-x^{2})^\frac{-1}{2}+(1-x^{2})^{\frac{1}{2}}$ $dy=-x^{2}(1-x^{2})^\frac{-1}{2}+(1-x^{2})^{\frac{1}{2}}\times dx$