Question

derivada de f(x)= (2x² + 3).(√x +2x)

Answer 1

$\mathrm{f(x)=(2x^2+3)(\sqrt{x}+2x)=(2x^2+3)(x^{\frac{1}{2}}+2x)}\\\\ \mathrm{*\ y=f(x).g(x)\ \to\ y'=f'(x).g(x)+g'(x).f(x)}\\\\ \mathrm{f'(x)=(2x^2+3)'(x^{\frac{1}{2}}+2x)+(x^{\frac{1}{2}}+2x)'(2x^2+3)}\\\\ \mathrm{f'(x)=(2.2x)(x^{\frac{1}{2}}+2x)+\bigg(\dfrac{1}{2}x^{-\frac{1}{2}}+2\bigg)(2x^2+3)}\\\\ \mathrm{f'(x)=4x.(x^{\frac{1}{2}}+2x)+\bigg(\dfrac{1}{2x^{\frac{1}{2}}}+2\bigg)(2x^2+3)}\\\\ \mathrm{f'(x)=4x^{\frac{3}{2}}+8x^2+\dfrac{2x^2+3}{2x^{\frac{1}{2}}}+4x^2+6}$
$\mathrm{f'(x)=4x^{\frac{3}{2}}+12x^2+x^{\frac{3}{2}}+\dfrac{3}{2x^{\frac{1}{2}}}+6}\\\\ \boxed{\mathbf{f'(x)=12x^2+5x\sqrt{x}+\dfrac{3}{2\sqrt{x}}+6}}$