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Question
Differentiate the following:
y = `(x^2 + 1) root(3)(x^2 + 2)`
Solution
y = `(x^2 + 1) root(3)(x^2 + 2)`
y = `(x^2 + 1) (x^2 + 2)^(1/3)`
`("d"y)/("d"x) = (x^2 + 1)^(1/3) (x^2 + 2)^(1/3- 1) (2x + 0) + (x^2 + 2)^(1/3) (2x + 0)`
`("d"y)/("d"x) = 1/3 (x^2 + 1) (x^2 + 2)^(- 2/3) xx 2x + (x^2 + 2)^(1/3) (2x)`
= `(2x(x^2 + 1))/(3(x^2 + 2)^(2/3)) + 2x (x^2 + 2)^(1/3)`
= `(2x(x^2 + 1) + 2x(x^2 + 2)^(1/3) * 3(x^2 + 2)^(2/3))/(3(x^2 + 2)^(2/3)`
= `(2x^3 + 2x + 6x (x^2 + 2)^(1/3 + 2/3))/(3(x^2 + 2)^(2/3)`
= `(2x^2 + 2x + 6x(x^2 + 2))/(3(x^2 + 2)^(2/3)`
= `(2x^3 + 2x + 6x^3 + 12x)/(3(x^2 + 2)^(2/3)`
`("d"y)/("d"x) = (8x^3 + 14x)/(3(x^2 + 2)^(2/3)`
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