Cho \(f\left( x \right) = {2.3^{{{\log }_{81}}x}} + 3\). Tính \(f'\left( 1 \right).\)

[Kim Thu Đoàn]

Cho \(f\left( x \right) = {2.3^{{{\log }_{81}}x}} + 3\). Tính \(f'\left( 1 \right).\)
A. \(f'\left( 1 \right) = 0\)
B. \(f'\left( 1 \right) = \frac{1}{2}\)
C. \(f'\left( 1 \right) = \frac{1}{4}\)
D. \(f'\left( 1 \right) = 2\)

 

[Nguyễn Hào]

Ta có: \(f\left( x \right) = {2.3^{{{\log }_{{3^4}}}x}} + 3 = {2.3^{\frac{1}{4}{{\log }_3}x}} + 3 = 2{\left( {{3^{{{\log }_3}x}}} \right)^{\frac{1}{4}}} + 3 = 2{x^{\frac{1}{4}}} + 3\)
\( \Rightarrow f'\left( x \right) = 2.\frac{1}{4}{x^{ - \frac{3}{4}}} = \frac{1}{2}{x^{ - \frac{3}{4}}} \Rightarrow f'\left( 1 \right) = \frac{1}{2}.\)