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1.
y' = arctan(x) + x*1/(1 + x²) - (1/2)[1/(1 + x²)]*2x
= arctan(x)
2.
y' = [(1 + √x)/(1 - √x)]*[(1 - √x)/(1 + √x)]'
= [(1 + √x)/(1 - √x)]*[(1 - √x)'(1 + √x) - (1 - √x)(1 + √x)'] /(1 + √x)²
= [(1 + √x)/(1 - √x)]*[-(1 + √x)/(2√x) - (1 - √x)/(2√x)]/(1 + √x)²
= [(1 + √x)/(1 - √x)]*(-1/√x)/(1 + √x)²
= -1/[(1 - x)√x]
y' = arctan(x) + x*1/(1 + x²) - (1/2)[1/(1 + x²)]*2x
= arctan(x)
2.
y' = [(1 + √x)/(1 - √x)]*[(1 - √x)/(1 + √x)]'
= [(1 + √x)/(1 - √x)]*[(1 - √x)'(1 + √x) - (1 - √x)(1 + √x)'] /(1 + √x)²
= [(1 + √x)/(1 - √x)]*[-(1 + √x)/(2√x) - (1 - √x)/(2√x)]/(1 + √x)²
= [(1 + √x)/(1 - √x)]*(-1/√x)/(1 + √x)²
= -1/[(1 - x)√x]
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