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Câu hỏi: Tìm đạo hàm của hàm số sau:
\(y = \left( {{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}.\)
\(y = \left( {{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}.\)
Phương pháp giải
Sử dụng công thức \(\left( {uvw} \right)' = u'vw + uv'w + uvw'\)
Lời giải chi tiết
Ta có:
\(\begin{array}{l}
y' = \left({{x^2} + 1} \right)'{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right)\left[ {{{\left({{x^3} + 1} \right)}^2}} \right]'{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}\left[ {{{\left({{x^4} + 1} \right)}^3}} \right]'\\
= 2x{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right)\left[ {2\left({{x^3} + 1} \right)\left({{x^3} + 1} \right)'} \right]{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}\left[ {3{{\left({{x^4} + 1} \right)}^2}\left({{x^4} + 1} \right)'} \right]\\
= 2x{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right)\left[ {2\left({{x^3} + 1} \right). 3{x^2}} \right]{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}\left[ {3{{\left({{x^4} + 1} \right)}^2}. 4{x^3}} \right]\\
= 2x{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ 6{x^2}\left({{x^2} + 1} \right)\left({{x^3} + 1} \right){\left({{x^4} + 1} \right)^3}\\
+ 12{x^3}\left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^2}
\end{array}\)
Sử dụng công thức \(\left( {uvw} \right)' = u'vw + uv'w + uvw'\)
Lời giải chi tiết
Ta có:
\(\begin{array}{l}
y' = \left({{x^2} + 1} \right)'{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right)\left[ {{{\left({{x^3} + 1} \right)}^2}} \right]'{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}\left[ {{{\left({{x^4} + 1} \right)}^3}} \right]'\\
= 2x{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right)\left[ {2\left({{x^3} + 1} \right)\left({{x^3} + 1} \right)'} \right]{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}\left[ {3{{\left({{x^4} + 1} \right)}^2}\left({{x^4} + 1} \right)'} \right]\\
= 2x{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right)\left[ {2\left({{x^3} + 1} \right). 3{x^2}} \right]{\left({{x^4} + 1} \right)^3}\\
+ \left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}\left[ {3{{\left({{x^4} + 1} \right)}^2}. 4{x^3}} \right]\\
= 2x{\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^3}\\
+ 6{x^2}\left({{x^2} + 1} \right)\left({{x^3} + 1} \right){\left({{x^4} + 1} \right)^3}\\
+ 12{x^3}\left({{x^2} + 1} \right){\left({{x^3} + 1} \right)^2}{\left({{x^4} + 1} \right)^2}
\end{array}\)