Question

Differentiate the following with respect to x.

`sqrtx + 1/root(3)(x) + e^x`

Answer 1

Let y = `sqrtx + 1/root(3)(x) + e^x`

y = `x^(1/2) + x^(1/3) + e^x`

`[because 1/root(3)(x) = 1/(x^(1/3)) = x^(1/3)]`

`"dy"/"dx" = "d"/"dx" (x^(1/2)) + "d"/"dx" (x^(-1/3)) + "d"/"dx" (e^x)`

`= 1/2 x^(1/2 - 1) + ((- 1)/3) x^((-1)/3 - 1) + e^x`

`= 1/2 x^((-1)/2) - 1/3 1/(x^(4/3)) + e^x`

`= 1/2 1/(x^(1/2)) - 1/3 1/(x^(4/3)) + e^x`

`= 1/(2sqrtx) - 1/(3 root(3)(x^4)) + e^x`