Home > Calculus > First Principles Example 2: x³

What is the second derivative of f(x)=ln (x+3)?

Answer 1

Scarlett Allison

#f''(x) = -1/(x+3)^2#

NOTE that the Derivative of #ln(x) = 1/x# and derivative of #1/x = -1/x^2#

#f'(x)=(d{ln(x+3)})/dx =1/(x+3)#

#f''(x) = (d{1/(x+3)})/dx = -1/(x+3)^2#

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Answer 2

Charles Anctil

#f''(x)=-1/(x+3)^2#

#"differentiate using the "color(blue)"chain rule"#

#"Given "y=f(g(x))" then"#

#dy/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"#

#rArrf'(x)=1/(x+3)xxd/dx(x+3)=1/(x+3)=(x+3)^-1#

#"differentiate "f'(x)" using the chain rule"#

#rArrf''(x)=-(x+3)^-2xxd/dx(x+3)=-1/(x+3)^2#

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Answer 3

Charles Anctil

The second derivative of ( f(x) = \ln(x+3) ) is:

[ f''(x) = -\frac{1}{(x+3)^2} ]

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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.