Derivatives of Inverse Hyperbolic Functions
This content covers the derivatives of inverse hyperbolic functions such as sinh^(-1)x, cosh^(-1)x, tanh^(-1)x, sech^(-1)x, cosech^(-1)x, coth^(-1)x. Learn how to find and derive these functions with detailed explanations and examples.
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Presentation Transcript
Derivatives of Inverse Hyperbolic functions We are familiar with hyperbolic functions. We learnt how to find the derivative of hyperbolic functions. Now we have to find the derivatives of Inverse hyperbolic functions i.e. Sinh-1x, cosh-1x, tanh-1x, sech-1x, cosech-1x, coth-1x these are inverse hyperbolic functions.
f Derivative of sinh 1? ???? =??? 1? ? =??? ? ?? =??? ? ?? ?? 1 = ?? ??? ? Since ??? 2? ??? 2? = 1
???2? = 1+???2? ??? ? = 1+??? 2? ?? ??= 1 1+??? 2? ?? ??= 1 1+?2 Where the sign of radical is same that of cosh? which we know cosh???always +?? ? ??(sinh 1?) = 1 1+?2
Derivative of tanh1?.[ ? < 1] Let y= tanh 1? ? = tanh ? ?? ?? = sech2? ?? 1 ?? = sec h2? Since sech2? + tanh2? = 1 sech2? = 1 tanh2? ?? 1 ?? = 1 tan h2? ?? 1 ??= ? ??(tanh 1?)= 1 ?2 1 1 ?2
Type equation here.Derivative of sech 1? Let y = sech 1? X = sech? ?? ?? = -sech?tanh? ?? ?? = 1 sech ? tanh ? We know that sech2? + tanh2? = 1 ??? 2y = 1-??? 2y tanh? = 1 ??? 2? ?? 1 ?? = sech ? 1 ??? 2?
= Where the sign of radical is same that of tanh y but we know that sech 1? is always +ve. tanh 1????????? + ??.
j Derivative of cosech 1? Let y = cosech 1? ? = cosech? ?? ?? = - cosech?coth? ?? ??= 1 cosech?coth? We know that ??? 2y ????? 2? = 1 ??? 2? = 1 + ????? 2? coth? = 1 + ????? 2? ?? ??= 1 cosechy1 +????? 2? 1 ? ??cosech 1? = |x| ?2+1 Type equation here.
Where the sign of radical is same that of coth y, coth y is +ve or ve according to value of x. Similarly we can find the derivative of cosh 1? &coth 1?
