Question

I need some help with this calculus 1 question

Answer 1

Explanation:

`f(x) = x \sqrt{9 - {x}^(2) } \\ \therefore \: f(x) = \sqrt{9 {x}^(2) - {x}^(4) } \\ \therefore \: f(x) = ({9 {x}^(2) - {x}^(4) } )^{ (1)/(2) } \\ differentiating \: both \: sides \: w.r.t. \: x \\ {f}^(')(x) = (1)/(2) * ({9 {x}^(2) - {x}^(4) } )^{ - (1)/(2) } \\ \hspace{30 pt} * (9 * 2x - 4 {x}^(3) ) \\ \therefore \: {f}^(')(x) = (1)/(2) * ({9 {x}^(2) - {x}^(4) } )^{ - (1)/(2) } \\\hspace{32 pt}* (18x - 4 {x}^(3) ) \\ \therefore \: {f}^(')(x) = (1)/(2) * ({9 {x}^(2) - {x}^(4) } )^{ - (1)/(2) } \\\hspace{32 pt} * 2(9x - 2 {x}^(3) ) \\ \therefore \: {f}^(')(x) = ({9 {x}^(2) - {x}^(4) } )^{ - (1)/(2) } * (9x - 2 {x}^(3) ) \\ \therefore \: {f}^(')(x) = \frac{1}{x({9 - {x}^(2) } )^{ (1)/(2) } } * x(9 - 2 {x}^(2) ) \\ \\ \: \: \: \: \: \: \red{ \boxed{ \bold{\therefore \: {f}^(')(x) = \frac{(9 - 2 {x}^(2) ) }{ \sqrt{{9 - {x}^(2) }}}}}}`