Question

derivada da função raiz de 3x¹-7

Answer 1

Regra da cadeia:

$\boxed{\boxed{\dfrac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)=\dfrac{dy}{du}\dfrac{du}{dx}}}$
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$f(x)=\sqrt{3x-7}$

Seja $h(x) = \sqrt{x}$ e $g(x)=3x-7$

Então

$h(g(x))=\sqrt{g(x)}=\sqrt{3x-7}~~~~\therefore~~~~\boxed{\boxed{f(x)=h(g(x))}}$

Portanto:

$f'(x)=h'(g(x))\cdot g'(x)\\\\\\f'(x)=\dfrac{1}{2\sqrt{3x-7}}\cdot\dfrac{d}{dx}(3x-7)\\\\\\f'(x)=\dfrac{1}{2\sqrt{3x-7}}\cdot3\\\\\\\boxed{\boxed{f'(x)=\dfrac{3}{2\sqrt{3x-7}}}}$