Question

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

Answer 1

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}`