Derivative of Inverse Trigonometric functions |
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It is important to note that all
inverse trigonometric functions are single-valued functions.
The domain and range of the
functions are restricted to certain intervals.
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Inverse sine function Domain: x
Î [-1, +1] Range: y
Î [-p/2, +p/2] y = sin-1x sin y = x
From the graph the slope
of y = sin-1 x is always positive. We take the positive root in the denominator of the
derivative formula. |
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Inverse cosine function Domain: x
Î [-1, +1] Range: y
Î [0, +p] y = cos-1x cos y = x
From the graph the slope
of y = cos-1 x is always negative. The derivative is negative. |
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Inverse tangent function Domain: x
Î (-¥, +¥) Range: y
Î [-p/2, +p/2] y = tan-1x tan y = x
From the graph the slope
of y = tan-1 x is always positive.
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Inverse cotangent function Domain: x
Î (-¥, +¥) Range: y
Î [-p/2, +p/2] y = cot-1x cot y = x
From the graph the slope
of y = cot-1 x is always negative.
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Inverse secant function Domain: x
Î ( -¥, +¥) / (-1, 1) Range: y
Î [0, +p]/{p/2} y = sec-1x sec y = x
From the graph the slope
of y = sec-1 x is always positive. We take the positive root in the denominator of the
derivative formula. |
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Inverse cosecant function Domain: x
Î ( -¥, +¥) / (-1, 1) Range: y
Î [-p/2, +p/2]/{0} y = csc-1x csc y = x
From the graph the slope
of y = csc-1 x is always negative.
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