# What are the first and second derivatives of f(x)=ln(-2e^(-6x^3)+x^2) ?

Jan 3, 2016

Chain, quotient, product rule. This has it all.

$f ' \left(x\right) = 2 \frac{18 {x}^{2} {e}^{- 6 {x}^{3}} + x}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}}$

$f ' ' \left(x\right) = \frac{2 x \cdot \left({e}^{6 {x}^{3}} + 18 x\right)}{{e}^{6 {x}^{3}} {x}^{2} - 2}$

If you actually need to see how the second derivative is done, leave a comment, note or pm.

#### Explanation:

$f \left(X\right) = \ln \left(- 2 {e}^{- 6 {x}^{3}} + {x}^{2}\right)$

$f ' \left(x\right) = \frac{1}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} \cdot \left(- 2 {e}^{- 6 {x}^{3}} + {x}^{2}\right) ' =$

$= \frac{\left(- 2 {e}^{- 6 {x}^{3}} + {x}^{2}\right) '}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} = \frac{\left(- 2 {e}^{- 6 {x}^{3}}\right) ' + \left({x}^{2}\right) '}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} =$

$= \frac{- 2 \left({e}^{- 6 {x}^{3}}\right) ' + \left({x}^{2}\right) '}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} = \frac{- 2 {e}^{- 6 {x}^{3}} \cdot \left(- 6 {x}^{3}\right) ' + 2 x}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} =$

$= \frac{- 2 {e}^{- 6 {x}^{3}} \cdot \left(- 6 \cdot 3 {x}^{2}\right) + 2 x}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} = \frac{36 {x}^{2} {e}^{- 6 {x}^{3}} + 2 x}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} =$

$= 2 \frac{18 {x}^{2} {e}^{- 6 {x}^{3}} + x}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}}$

Since for the second derivative we will use ${e}^{- 6 {x}^{3}}$ several times, let's remember that its derivative is $- 18 {x}^{2} {e}^{- 6 {x}^{3}}$

$f ' ' \left(x\right) = 2 \frac{\left(18 {x}^{2} {e}^{- 6 {x}^{3}} + x\right) ' \left(- 2 {e}^{- 6 {x}^{3}} + {x}^{2}\right) - \left(18 {x}^{2} {e}^{- 6 {x}^{3}} + x\right) \left(- 2 {e}^{- 6 {x}^{3}} + {x}^{2}\right) '}{- 2 {e}^{- 6 {x}^{3}} + {x}^{2}} ^ 2$

I'm sorry, but I just realised that whoever put this assignment to you doesn't really like you. Since the function is too big for Socratic's format already and not even half of the exercise is done, I will just post the derivative here:

$f ' ' \left(x\right) = \frac{2 x \cdot \left({e}^{6 {x}^{3}} + 18 x\right)}{{e}^{6 {x}^{3}} {x}^{2} - 2}$