DerivativE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Question

If h(x) = tan(3x) / x^2 , find the derivative of h(x).

Answer 1

The function h (x) = tan(3x) / x² .

h (x) = tan(3x) * x^{- 2} .

Differentiate the function with respect to x.

h ' (x) = [ tan(3x) * x^{- 2} ] '.

Apply product rule : (uv) ' = uv ' + vu ' .

h ' (x) = tan(3x) * (x^{- 2}) ' + x^{- 2}(tan 3x) '

Apply formulas : (xⁿ) ' = nx^{n - 1}

                          (tan x) ' = sec² x.

h ' (x) = - 2tan(3x)(x^{- 3}) + 3x^{- 2}(sec² 3x)

h ' (x) = [ - 2tan(3x)/(x³) ] + [ 3(sec² 3x)/x² ]

h ' (x) = [ 3x(sec² 3x) - 2tan(3x)] / x³ .

The derivative of h(x) is [ 3x(sec² 3x) - 2tan(3x)] / x³ .