Question

Use logarithmic differentiation to differentiate the question below


`y = x \sqrt[3]{1 + {x}^(2) }`
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Answer 1

`\orange{ \bold{(dy)/(dx) =\frac{ 5{x}^(2) + 3 }{3\sqrt[3]{(1 + {x}^(2))^(2) } }}}`

Explanation:

`y = x \sqrt[3]{1 + {x}^(2) } \\ assuming \: log \: both \: sides \\log y = log(x \sqrt[3]{1 + {x}^(2) } ) \\ \therefore log y = logx + log(\sqrt[3]{1 + {x}^(2) } ) \\ \therefore log y = logx + (1)/(3) log({1 + {x}^(2) } ) \\ differentiating \: both \: sides \: w.r.t.x \\ (1)/(y) (dy)/(dx) = (1)/(x) + (1)/(3) . \frac{1}{(1 + {x}^(2)) } (0 + 2x) \\ (1)/(y) (dy)/(dx) = (1)/(x) + \frac{2x}{3(1 + {x}^(2)) }\\ (1)/(y) (dy)/(dx) =\frac{3(1 + {x}^(2)) + 2 {x}^(2) }{3x(1 + {x}^(2)) }\\ (1)/(y) (dy)/(dx) =\frac{3 + 3{x}^(2) + 2 {x}^(2) }{3x(1 + {x}^(2)) }\\ (1)/(y) (dy)/(dx) =\frac{3 + 5{x}^(2) }{3x(1 + {x}^(2)) }\\ (dy)/(dx) =\frac{y(3 + 5{x}^(2) )}{3x(1 + {x}^(2)) } \\ \\ (dy)/(dx) =\frac{x \sqrt[3]{1 + {x}^(2) } (3 + 5{x}^(2) )}{3x(1 + {x}^(2)) }\\ \\ (dy)/(dx) =\frac{(3 + 5{x}^(2) )\sqrt[3]{1 + {x}^(2) } }{3(1 + {x}^(2)) }\\ \\ \purple{ \bold{(dy)/(dx) =\frac{ 5{x}^(2) + 3 }{3\sqrt[3]{(1 + {x}^(2))^(2) } }}}`