Nội dung bài giảng
Tìm đạo hàm của mỗi hàm số sau đến cấp được cho kèm theo.
a. \(f\left( x \right) = {x^4} - \cos 2x,{f^{\left( 4 \right)}}\left( x \right)\)
b. \(f\left( x \right) = {\cos ^2}x,{f^{\left( 5 \right)}}\left( x \right)\)
c. \(f\left( x \right) = {\left( {x + 10} \right)^6},{f^{\left( n \right)}}\left( x \right)\)
Giải:
a. Ta có:
\(\begin{array}{l}
f'\left( x \right) = 4{x^3} + 2\sin 2x\\
f"\left( x \right) = 12{x^2} + 4\cos 2x\\
{f^{\left( 3 \right)}} = 24x - 8\sin 2x\\
{f^{\left( 4 \right)}}\left( x \right) = 24 - 16\cos 2x
\end{array}\)
b.
\(\begin{array}{l}
f'\left( x \right) = 2\cos x\left( { - \sin x} \right) = - \sin 2x\\
f"\left( x \right) = - 2\cos 2x\\
{f^{\left( 3 \right)}}\left( x \right) = 4\sin 2x\\
{f^{\left( 4 \right)}} = 8\cos 2x\\
{f^{\left( 5 \right)}}\left( x \right) = - 16\sin 2x
\end{array}\)
c.
\(\begin{array}{l}
f'\left( x \right) = 6{\left( {x + 10} \right)^5}\\
f"\left( x \right) = 30{\left( {x + 10} \right)^4}\\
{f^{\left( 3 \right)}}\left( x \right) = 120{\left( {x + 10} \right)^3}\\
{f^{\left( 4 \right)}}\left( x \right) = 360{\left( {x + 10} \right)^2}\\
{f^{\left( 5 \right)}}\left( x \right) = 720\left( {x + 10} \right)\\
{f^{\left( 6 \right)}}\left( x \right) = 720\\
{f^{\left( n \right)}}\left( x \right) = 0,\forall n \ge 7
\end{array}\)