Question

Differentiate the following w.r.t. x:

`e^x + e^(x^2) +... + e^(x^3)`

Answer 1

Let y = `e^x + e^(x^2) +... + e^(x^5)`

Differentiating both sides with respect to x,

`dy/dx = d/dx (e^x) + d/dx (e^(x^2)) = d/dx (e^(x^3)) + d/dx (e^(x^4)) + d/dx (e^(x^5))`

`= e^x + e^(x^2) d/dx (x^2) + e^(x^3) d/dx (x^3) + e^(x^4) d/dx (x^4) + e^(x^5) d/dx x^5 `

`= e^x + e^(x^2). 2 x + e^(x^3). 3x^2 + e^(x^4) .4x^3 + e^(x^5). 5x^4`

`= e^x + 2xe^(x^2) + 3x^2 e^(x^3) + 4x^3 e^(x^4) + 5x^4 e^(x^5)`

`=e^x (1 + 2xe^x + 3x^2 e^(x^2) + 4x^3 e^(x^3) + 5x^4 e^(x^4))`