`y = 2xsinh^-1(2x) - sqrt(1+4x^2)` Find the derivative of the function

Expert Answers

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The expression of this function includes difference, product and two table functions (except the polynomials), `sinh^(-1)(z)` and `sqrt(z).`

The difference rule is `(u-v)' = u' - v',` the product rule is `(uv)' = u'v + uv',` the chain rule is `(u(v(x)))' = u'(v(x))*v'(x).`

The derivative of `sinh^(-1)(z)`  is `1/sqrt(1 + z^2),` the derivative of `sqrt(z)` is `1/(2sqrt(z)).`

These rules together give us

`y' = 2sinh^-1(2x) + 2x ((2x)')/sqrt(1+(2x)^2) - ((4x^2)')/(2sqrt(1+(2x)^2)) =`

`= 2sinh^-1(2x) + 2x (2)/sqrt(1+4x^2) - (8x)/(2sqrt(1+4x^2)) =`

`= 2sinh^-1(2x).`


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