Y = Tanh^-1(sqrt(x)) Find The Derivative Of The Function - Blogger

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Monday, September 22, 2014

y = tanh^-1(sqrt(x)) Find the derivative of the function

Derivative of a function f with respect to x is denoted as f'(x) or  y' .To solve for derivative of y or (y') for the given problem: y = tanh^(-1)(sqrt(x)) , we follow the basic derivative formula for inverse hyperbolic function:d/(dx)(tanh^(-1)(u))= ((du)/(dx))/(1-u^2) where |u|lt1 .Let: u =sqrt(x)Apply the Law of Exponent: sqrt(x) = x^(1/2)Solve for the derivative of u using the Power Rule for derivative: d/(dx)x^n=n*x^(n+1) * d(x)Then,du=1/2x^(1/2-1)*1dxdu=1/2x^(-1/2) dxApply the Law of Exponent:x^(-n)= 1/x^n. du=1/(2x^(1/2)) dx Rearrange into:(du)/(dx)=1/(2x^(1/2))(du)/(dx)=1/(2sqrt(x))      Apply the derivative formula, we get:d/(dx)(tanh^(-1)(sqrt(x)))= ((1/(2sqrt(x))))/((1-(sqrt(x))^2))                               =((1/(2sqrt(x))))/((1-x))                               =(1/(2sqrt(x)))*1/((1-x))                               =1/(2sqrt(x)(1-x))Final answer:d/(dx)(tanh^(-1)(sqrt(x)))=1/(2sqrt(x)(1-x))

- September 22, 2014

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