Question

Evaluate the following functions in differential operator form.

1. D2((x+1)2/x-1) Answer: 8/(x-1)³
2. D(ln(x²+2x+1) Answer: 2/x-1

Answer 1

Q1.

Let y = $\frac{(x+1)²}{x-1}$

So y = $\frac{(x-1)²+4x}{x-1}= \frac{(x-1)²+4(x-1)+4}{x-1}$

=> y = (x-1) +4 + $\frac{4}{x-1}$

=> y = x + 3 + $\frac{4}{x-1}$

So $\frac{dy}{dx}= 1+0-\frac{4}{(x-1)²}$

Differentiating again

$\frac{d²y}{dx²}= 0-\frac{4*(-2)}{(x-1)³}= \frac{8}{(x-1)³}$

Therefore D²($\frac{(x+1)²}{x-1}$ ) = $\frac{8}{(x-1)³}$

Q2. D(ln(x²+2x+1))

= D(ln(x+1)²)

= D(2ln(x+1)) since ln(A)ⁿ = n ln(A)

= 2D(ln(x+1))

= $2\frac{d}{dx}(ln(x+1))$

= $\frac{2}{x+1}$

Note: Given answer of question number 2 is wrong.