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Câu hỏi: Đạo hàm của hàm số $f\left( x \right)=\dfrac{{{3}^{x}}-1}{{{3}^{x}}+1}$ là:
A. ${f}'\left( x \right)=-\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}$.
B. ${f}'\left( x \right)=\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}$.
C. ${f}'\left( x \right)=-\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}\ln 3$.
D. ${f}'\left( x \right)=\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}\ln 3$.
A. ${f}'\left( x \right)=-\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}$.
B. ${f}'\left( x \right)=\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}$.
C. ${f}'\left( x \right)=-\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}\ln 3$.
D. ${f}'\left( x \right)=\dfrac{2}{{{\left( {{3}^{x}}+1 \right)}^{2}}}{{.3}^{x}}\ln 3$.
${{\left( \dfrac{u}{v} \right)}^{\prime }}=\dfrac{{u}'.v-u.{v}'}{{{v}^{2}}}$ ; ${{\left( {{a}^{x}} \right)}^{\prime }}={{a}^{x}}.\ln a$
${{\left( \dfrac{{{3}^{x}}-1}{{{3}^{x}}+1} \right)}^{\prime }}=\dfrac{{{\left( {{3}^{x}}-1 \right)}^{\prime }}.\left( {{3}^{x}}+1 \right)-\left( {{3}^{x}}-1 \right){{\left( {{3}^{x}}+1 \right)}^{\prime }}}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$
$=\dfrac{{{3}^{x}}.\ln 3.\left( {{3}^{x}}+1 \right)-\left( {{3}^{x}}-1 \right){{.3}^{x}}.\ln 3}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$
$=\dfrac{{{3}^{x}}.\ln {{3.3}^{x}}+{{3}^{x}}.\ln 3-{{3}^{x}}{{.3}^{x}}.\ln 3+{{3}^{x}}.\ln 3}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$
$=2.\dfrac{{{3}^{x}}.\ln 3}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$.
${{\left( \dfrac{{{3}^{x}}-1}{{{3}^{x}}+1} \right)}^{\prime }}=\dfrac{{{\left( {{3}^{x}}-1 \right)}^{\prime }}.\left( {{3}^{x}}+1 \right)-\left( {{3}^{x}}-1 \right){{\left( {{3}^{x}}+1 \right)}^{\prime }}}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$
$=\dfrac{{{3}^{x}}.\ln 3.\left( {{3}^{x}}+1 \right)-\left( {{3}^{x}}-1 \right){{.3}^{x}}.\ln 3}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$
$=\dfrac{{{3}^{x}}.\ln {{3.3}^{x}}+{{3}^{x}}.\ln 3-{{3}^{x}}{{.3}^{x}}.\ln 3+{{3}^{x}}.\ln 3}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$
$=2.\dfrac{{{3}^{x}}.\ln 3}{{{\left( {{3}^{x}}+1 \right)}^{2}}}$.
Đáp án C.