# Derive the derivative of (x^2 + 3x)^2 from first principles.

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The derivative of f(x) = (x^2 + 3x)^2 has to be derived from first principles.

`f'(x) = lim_(h->0) (f(x+h) - f(x))/h`

=> `lim_(h->0) (((x+h)^2 + 3(x+h))^2 - (x^2 + 3x)^2)/h`

=> `lim_(h->0) ((x^2 + 2xh + h^2 + 3x + 3h)^2 + - x^4 - 6x^3 - 9x^2)/h`

=> `lim_(h->0) (x^4 + 4x^2h^2 + h^4 + 9x^2 + 9h^2 + 4x^3h + 2x^2h^2 + 6x^3 + 6x^2h)/h `

`+ (4xh^3 + 12x^2h + 12xh^2 + 6h^2x + 6h^3 + 18xh - x^4 - 6x^3 - 9x^2)/h`

=>

`lim_(h->0) ( 4x^2h^2 + h^4 + 9h^2 + 4x^3h + 2x^2h^2)/h `

`+ (6x^2h + 4xh^3 + 12x^2h + 12xh^2 + 6h^2x + 6h^3 + 18xh)/h`

`lim_(h->0) 4x^2h + h^3 + 9h + 4x^3 + 2x^2h+6x^2 + 4xh^2 `

`+ 12x^2 + 12xh + 6hx + 6h^2 + 18x`

substitute h = 0

=> 4x^3 + 6x^2 + 12x^2 + 18x

=> 4x^3 + 18x^2 + 18x

**The required derivative of (x^2 + 3x)^2 is 4x^3 + 18x^2 + 18x**