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ddxsin(x)=cos(x)ddxtan(x)=sec2(x)ddxsec(x)=sec(x)⋅tan(x) |
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ddxcos(x)=−sin(x)ddxcot(x)=−csc2(x)ddxcsc(x)=−csc(x)⋅cot(x) |
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ddxsin−1(x)=1√1−x2ddxtan−1(x)=11+x2ddxsec−1(x)=1x⋅√x2−1 |
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ddxcos−1(x)=−1√1−x2ddxcot−1(x)=−11+x2ddxcsc−1(x)=−1x⋅√x2−1 |
Take the last trig function as an example. Let f(x)=csc−1(x),
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f(csc(x))=csc−1(csc(x))=xf′(csc(x))⋅csc′(x)=1f′(csc(x))=1csc′(x)=−1csc(x)⋅cot(x) |
Since sin2(x)+cos2(x)=1, divide both sides by sin2(x):
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1+cot2(x)=csc2(x)cot(x)=√csc2(x)−1 |
From above,
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f′(csc(x))=−1csc(x)⋅cot(x)f′(csc(x))=−1csc(x)⋅√csc2(x)−1f′(x)=−1x⋅√x2−1∴ |
# posted by rot13(Unafba Pune) @ 10:48 PM
