Home > Calculus > Differentiating Logarithmic Functions with Base e

How do you find the second derivative of # ln^2 (x)#?

Answer 1

#:. (d^2y)/(dx^2)={2(1-lnx)}/x^2=ln{(e/x)^(2/x^2)}.#

Let #y=ln^2(x)=(lnx)^2#.

#:. dy/dx=2lnx*d/dx(lnx)#.......[Chain Rule].....#=(2lnx)/x.#

#:. (d^2y)/(dx^2)=d/dx{dy/dx}#........................[Defn.] #=d/dx{(2lnx)/x}# #=2[{x*d/dx(lnx)-(lnx)*d/dx(x)}/x^2]#..........[Quotient Rule] #=2[{x*1/x-(lnx)(1)}/x^2]# #={2(1-lnx)}/x^2# #=(2/x^2)(lne-lnx)# #=(2/x^2)*ln(e/x)# #=ln{(e/x)^(2/x^2)}#

Maths. is amazing, enjoy it!

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Scarlett Allison

To find the second derivative of ( \ln^2(x) ), first differentiate the function ( \ln^2(x) ) with respect to ( x ) to find the first derivative, then differentiate the first derivative again with respect to ( x ) to find the second derivative.

The result is:

[ \frac{d^2}{dx^2}[\ln^2(x)] = \frac{2}{x^2} - \frac{2}{x^3} ]

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.