求高等数学高阶导数问题 30
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f(x) = x^2/(1-x^2) = (x^2-1+1)/(1-x^2)
= -1+(1/2)[1/(1+x)+1/(1-x)] = -1+(1/2)[1/(x+1)-1/(x-1)]
f'(x) = (1/2)[-1/(x+1)^2 - (-1)/(x-1)^2]
f''(x) = (1/2)[(-1)(-2)/(x+1)^3 - (-1)(-2)/(x-1)^3]
f'''(x) = (1/2)[(-1)(-2)(-3)/(x+1)^4 - (-1)(-2)(-3)/(x-1)^4]
...................................................................................
f^(n)(x) = (1/2)(-1)^n n! [1/(x+1)^(n+1) - 1/(x-1)^(n+1)]
f^(n)(0) = (1/2)(-1)^n n! [1 - 1] = 0
= -1+(1/2)[1/(1+x)+1/(1-x)] = -1+(1/2)[1/(x+1)-1/(x-1)]
f'(x) = (1/2)[-1/(x+1)^2 - (-1)/(x-1)^2]
f''(x) = (1/2)[(-1)(-2)/(x+1)^3 - (-1)(-2)/(x-1)^3]
f'''(x) = (1/2)[(-1)(-2)(-3)/(x+1)^4 - (-1)(-2)(-3)/(x-1)^4]
...................................................................................
f^(n)(x) = (1/2)(-1)^n n! [1/(x+1)^(n+1) - 1/(x-1)^(n+1)]
f^(n)(0) = (1/2)(-1)^n n! [1 - 1] = 0
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