How do you differentiate #f(x)=e^sqrt((3x^2)/(2x-3)) # using the chain rule?

1 Answer
t0hierry
Aug 11, 2017

We are going to use the simple rule of the derivative of a power function #df(x)^n/dx = n f^{n-1}df(x)/dx# to find, posing

#g(x) = \sqrt{\frac{3x^2}{2x-3}} = [\frac{3x^2}{2x-3}]^\frac{1}{2}#
and
#h(x) = e^g(x)#
that
#\frac{dh}{dx} = \frac{dg}{dx} h#
#\frac{dh}{dx} = 1/2 [\frac{3x^2}{2x-3}]^\frac{-1}{2]\frac{(2x-3)6x - 3x^2(2)}{(2x-3)^2}#
where I have used the chain rule for the ratio of two functions
#\frac{dh}{dx} = \frac{6x^2-9x - 3x^2}([(2x-3)^2]}[\frac{3x^2}{2x-3}]^\frac{-1}{2}#
#\frac{dh}{dx} = \frac{3x^2 - 9x}{[3x^2]^\frac{1}{2}}(2x-3)^\frac{-5}{2}#
#\frac{dh}{dx} = sqrt{3}(x -3)(2x-3)^\frac{-5}{2}#

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