Home > Calculus > Relationship between First and Second Derivatives of a Function

What is the second derivative of #f(x)= (x+1)/(x-1)#?

Answer 1

#\frac{d^2}{dx^2}(f(x))=4/(x-1)^3#

Given #f(x)=(x+1)/(x-1)# Assuming you know quotient rule, Differentiating once, #\frac{d}{dx}(f(x))=((x-1)-(x+1))/(x-1)^2=(-2)/(x-1)^2#

Differentiating again, we get #\frac{d^2}{dx^2}(f(x))=4/(x-1)^3#

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

Answer 2

Christopher Agresta

To find the second derivative of ( f(x) = \frac{x+1}{x-1} ), follow these steps:

Find the first derivative of ( f(x) ) using the quotient rule. [ f'(x) = \frac{(x-1)(1) - (x+1)(1)}{(x-1)^2} ] [ f'(x) = \frac{x-1 - x - 1}{(x-1)^2} ] [ f'(x) = \frac{-2}{(x-1)^2} ]
Now, differentiate ( f'(x) ) with respect to ( x ) to find the second derivative. [ f''(x) = \frac{d}{dx}\left(\frac{-2}{(x-1)^2}\right) ]
Apply the chain rule and the power rule for differentiation. [ f''(x) = \frac{2 \cdot 2(x-1)^{2-1} \cdot (1)}{(x-1)^{2 \cdot 1}} ] [ f''(x) = \frac{4(x-1)}{(x-1)^2} ]
Simplify the expression. [ f''(x) = \frac{4(x-1)}{(x-1)^2} ] [ f''(x) = \frac{4}{x-1} ]

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.