Question

Dada a função
f(x) = log_{a}x = \frac{lnx}{lna}, marque a alternativa correta que apresenta a derivada f'(x).

a) x/ ln a.
b) ln a.
c) a/ ln x
d) 1/x. ln a
e) u. ln a

Answer 1

$f(x)=\mathrm{\ell og}_{a\,}x=\dfrac{\mathrm{\ell n\,}x}{\mathrm{\ell n\,}a}\\ \\ \\ f(x)=\dfrac{1}{\mathrm{\ell n\,}a}\cdot \mathrm{\ell n\,}x$


Derivando a expressão acima, temos

$f'(x)=\left(\dfrac{1}{\mathrm{\ell n\,}a}\cdot \mathrm{\ell n\,}x \right )'\\ \\ \\ f'(x)=\dfrac{1}{\mathrm{\ell n\,}a}\cdot \left(\mathrm{\ell n\,}x \right )'\\ \\ \\ f'(x)=\dfrac{1}{\mathrm{\ell n\,}a}\cdot \dfrac{1}{x}\\ \\ \\ f'(x)=\dfrac{1}{x\mathrm{\,\ell n\,}a}$


Resposta: alternativa $\text{d) }\dfrac{1}{x\mathrm{\,\ell n\,}a}.$