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Câu 47 trang 219 SGK Đại số và Giải tích 11 Nâng cao

a. Cho hàm số
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LG a

Cho hàm số \(f\left( x \right) = \tan x.\) Tính \({f^{\left( n \right)}}\left( x \right)\) với n = 1, 2, 3.

Phương pháp giải:

Sử dụng công thức đạo hàm \(\left( {\tan x} \right)' = 1 + {\tan ^2}x\)

Lời giải chi tiết:

f’(x) = 1 + tan2x

f’’(x) = 2tanx(1 + tan2x) = 2tanx + 2tan3x

f(3)(x) = 2(1 + tan2x) + 2.3tan2x(1 + tan2x)

= 2+ 2tan2x + 6tan2x+ 6tan4x

= 2+ 8tan2x+ 6tan4x

LG b

Chứng minh rằng nếu \(f\left( x \right) = {\sin ^2}x\) thì \({f^{\left( {4n} \right)}}\left( x \right) =  - {2^{4n - 1}}\cos 2x\)

Phương pháp giải:

Chứng minh bằng phương pháp qui nạp.

Lời giải chi tiết:

\({f^{\left( {4n} \right)}}\left( x \right) =  - {2^{4n - 1}}\cos 2x\)  (1) Với n = 1 ta có: 

\(\begin{array}{l}
f'\left( x \right) = 2\sin x\cos x= \sin 2x\\
f"\left( x \right) = 2\cos 2x\\
{f^{\left( 3 \right)}}\left( x \right) = - 4\sin 2x\\
{f^{\left( 4 \right)}}\left( x \right) = - 8\cos 2x =  - {2^{4.1 - 1}}\cos 2x
\end{array}\)

Vậy (1) đúng với n = 1 Giả sử (1) đúng với n = k tức là :  \({f^{\left( {4k} \right)}}\left( x \right) =  - {2^{4k - 1}}\cos 2x\) Với n = k + 1 ta có : 

\(\begin{array}{l}
{f^{\left( {4k + 1} \right)}}\left( x \right) = \left( {{f^{\left( {4k} \right)}}\left( x \right)} \right)' = {2^{4k}}\sin 2x\\
{f^{\left( {4k + 2} \right)}}\left( x \right) = {2^{4k + 1}}\cos 2x\\
{f^{\left( {4k + 3} \right)}}\left( x \right) = - {2^{4k + 2}}\sin 2x\\
{f^{\left( {4k + 4} \right)}}\left( x \right) = - {2^{4k + 3}}\cos 2x \\=  - {2^{4\left( {k + 1} \right) - 1}}\cos 2x
\end{array}\)

Vậy (1) đúng với n = k + 1 do đó (1) đúng với mọi n.

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