What is the derivative of y=(7e^(3x^2+5x))(x^1/2)?

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## ` `

`y = 7e^(3x^2+5x) xx sqrt(x)`

We can use the product rule to differentiate this.

`(uv)' = u'v+ uv'`

Let `u = e^(3x^2+5x)`

let `t = (3x^2+5x)`

Then by chain rule,

`(du)/(dx) = (du)/(dt) xx (dt)/(dx)`

`(du)/(dx) = e^t xx (6x+5)`

Therfore,

`(dy)/(dx) = 7[e^(3x^2+5x) xx (6x+5) xx sqrt(x) + e^(3x^2+5x) xx (1/2 xx x^(-1/2))]`

`(dy)/(dx) = 7/2[2e^(3x^2+5x) xx (6x+5) xx sqrt(x) + e^(3x^2+5x)/sqrt(x)]`

This gives,

`(dy)/(dx) = 7/2e^(3x^2+5x)[2(6x+5) xx sqrt(x) + 1/sqrt(x)]`

`(dy)/(dx) = 7/2e^(3x^2+5x)[(2(6x+5) xx x + 1)/sqrt(x)]`

`(dy)/(dx) = 7/2e^(3x^2+5x) xx ((12x^2+5x+1))/sqrt(x)`

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