Question

calcule a segunda derivada da função f(x)=1/x-log x e determine o valor desta segunda derivada no ponto x=5

Answer 1

$f\left(x \right )=\dfrac{1}{x}-\mathrm{\ell og\,}x\\ \\ \\ f'\left(x \right )=\left(\dfrac{1}{x}-\mathrm{\ell og\,}x \right )'\\ \\ f'\left(x \right )=\left(x^{-1}-\dfrac{\mathrm{\ell n\,}x}{\mathrm{\ell n\,}10} \right )'\\ \\ f'\left(x \right )=-1\cdot x^{-1-1}-\dfrac{1}{\mathrm{\ell n\,}10}\cdot \dfrac{1}{x}\\ \\ f'\left(x \right )=-x^{-2}-\dfrac{1}{x\mathrm{\,\ell n\,}10}\\ \\ \\ f''\left(x \right )=\left(-x^{-2}-\dfrac{1}{x\mathrm{\,\ell n\,}10} \right )'\\ \\ f''\left(x \right )=\left(-x^{-2}-\dfrac{1}{\mathrm{\ell n\,}10}\cdot x^{-1} \right )'\\ \\ f''\left(x \right )=-\left(-2 \right )x^{-2-1}-\dfrac{\left(-1\right)}{\mathrm{\ell n\,}10}\cdot x^{-1-1}\\ \\ f''\left(x \right )=2x^{-3}+\dfrac{x^{-2}}{\mathrm{\ell n\,}10}\\ \\ \boxed{f''\left(x \right )=\dfrac{2}{x^{3}}+\dfrac{1}{x^{2}\mathrm{\,\ell n\,}10}}\\ \\ \\ f''\left(5 \right )=\dfrac{2}{\left(5 \right )^{3}}+\dfrac{1}{\left(5 \right )^{2}\mathrm{\,\ell n\,}10}$

$\boxed{f''\left(5 \right )=\dfrac{2}{125}+\dfrac{1}{25\mathrm{\,\ell n\,}10}}$