What is the second derivative of f(x)= 2x^3- sqrt(4x-5)?

Jan 10, 2016

$12 x + \frac{4}{4 x - 5} ^ \left(\frac{3}{2}\right)$

Explanation:

to obtain the second derivative means differentiating the function to obtain f'(x) and then differentiating f'(x) to obtain f''(x).

rewriting f(x) as $f \left(x\right) = 2 {x}^{3} - {\left(4 x - 5\right)}^{\frac{1}{2}}$

applying the chain rule :

$f ' \left(x\right) = 6 {x}^{2} - \frac{1}{2} {\left(4 x - 5\right)}^{- \frac{1}{2}} . \frac{d}{\mathrm{dx}} \left(4 x - 5\right)$

$f ' \left(x\right) = 6 {x}^{2} - \frac{1}{2} \left(4 x - 5\right) .4 = 6 {x}^{2} - 2 {\left(4 x - 5\right)}^{- \frac{1}{2}}$

$f ' ' \left(x\right) = 12 x + {\left(4 x - 5\right)}^{- \frac{3}{2}} . \frac{d}{\mathrm{dx}} \left(4 x - 5\right)$

$\Rightarrow f ' ' \left(x\right) = 12 x + {\left(4 x - 5\right)}^{- \frac{3}{2}} .4 = 12 x + \frac{4}{4 x - 5} ^ \left(\frac{3}{2}\right)$