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

Question

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

Elijah Abram · Accepted Answer

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

#f'(x)=2(18x^2e^(-6x^3)+x)/(-2e^(-6x^3)+x^2)#

#f''(x)=(2x*(e^(6x^3)+18x))/(e^(6x^3)x^2-2)#

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

#f(X)=ln(-2e^(-6x^3)+x^2)# #f'(x)=1/(-2e^(-6x^3)+x^2)*(-2e^(-6x^3)+x^2)'=# #=((-2e^(-6x^3)+x^2)')/(-2e^(-6x^3)+x^2)=((-2e^(-6x^3))'+(x^2)')/(-2e^(-6x^3)+x^2)=# #=(-2(e^(-6x^3))'+(x^2)')/(-2e^(-6x^3)+x^2)=(-2e^(-6x^3)*(-6x^3)'+2x)/(-2e^(-6x^3)+x^2)=# #=(-2e^(-6x^3)*(-6*3x^2)+2x)/(-2e^(-6x^3)+x^2)=(36x^2e^(-6x^3)+2x)/(-2e^(-6x^3)+x^2)=# #=2(18x^2e^(-6x^3)+x)/(-2e^(-6x^3)+x^2)# Since for the second derivative we will use #e^(-6x^3)# several times, let's remember that its derivative is #-18x^2e^(-6x^3)# #f''(x)=2((18x^2e^(-6x^3)+x)'(-2e^(-6x^3)+x^2)-(18x^2e^(-6x^3)+x)(-2e^(-6x^3)+x^2)')/(-2e^(-6x^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''(x)=(2x*(e^(6x^3)+18x))/(e^(6x^3)x^2-2)#

Evelyn Agard · Answer

undefined The first derivative of the function $ f(x) = \ln(-2e^{-6x^3} + x^2) $ can be found using the chain rule and the derivative of the natural logarithm function. The second derivative can be obtained by differentiating the first derivative.

Here are the steps:

1. Find the first derivative $ f'(x) $.
2. Find the second derivative $ f''(x) $.

This will yield the first and second derivatives of the given function.

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