Question

calcular derivada
f(x) =3 sen x (vezes) log 3 x

Answer 1

Regra do produto:

$\boxed{\boxed{\dfrac{d}{dx}[f(x)\cdot g(x)]=f'(x)\cdot g(x)+g'(x)\cdot f(x)}}$

Da regra do produto, tiramos

$\boxed{\boxed{\dfrac{d}{dx}kf(x)=kf'(x)~~~(k~\'e~uma~constante~real~qualquer)}}$

Derivada do seno:

$\boxed{\boxed{\dfrac{d}{dx}sen(x)=cos(x)}}$

Derivada de logaritmos (em qualquer base permitida):

$\boxed{\boxed{\dfrac{d}{dx}log_{a}(x)=\dfrac{d}{dx}\left(\dfrac{ln(x)}{ln(a)}\right)=\dfrac{1}{x\cdot ln(a)}}}$
______________________________

$f(x)=3sen(x)\cdot log_{3}(x)\\\\\\f'(x)=log_{3}(x)\cdot\dfrac{d}{dx}[3sen(x)]+3sen(x)\cdot\dfrac{d}{dx}[log_{3}(x)]\\\\\\f'(x)=log_{3}(x)\cdot3\dfrac{d}{dx}[sen(x)]+3sen(x)\cdot\dfrac{1}{x\cdot ln(3)}\\\\\\\boxed{\boxed{f'(x)=3\cdot log_{3}(x)\cdot cos(x)+\dfrac{3\cdot sen(x)}{x\cdot ln(3)}}}$

Podemos simplificar a expressão:

$f'(x)=3\cdot \dfrac{ln(x)}{ln(3)}\cdot cos(x)+\dfrac{3\cdot sen(x)}{x\cdot ln(3)}\\\\\\f'(x)=\dfrac{3x\cdot ln(x)\cdot cos(x)}{x\cdot ln(3)}+\dfrac{3\cdot sen(x)}{x\cdot ln(3)}\\\\\\f'(x)=\dfrac{3x\cdot ln(x)\cdot cos(x)+3\cdot sen(x)}{x\cdot ln(3)}\\\\\\\boxed{\boxed{f'(x)=\dfrac{3\cdot(x\cdot ln(x)\cdot cos(x)+sen(x))}{x\cdot ln(3)}}}$