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a_(n)=(n^(2)-2n+1)/(n-1)

$33$ . $a_{n}=\frac{n^{2}-2 n+1}{n-1}$

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Find derivatives using the chain rule I

Full solution

Q.

33

.

a_{n}=\frac{n^{2}-2 n+1}{n-1}

Identify function: Identify the function to differentiate. The function is $a_{n}=\frac{n^{2}-2n+1}{n-1}$ .
Use quotient rule: Recognize that this is a quotient, so we'll use the quotient rule: $(v*u' - u*v') / v^2$ , where $u=n^2-2n+1$ and $v=n-1$ .
Differentiate numerator: Differentiate the numerator $u=n^2-2n+1$ to get $u'=2n-2$ .
Differentiate denominator: Differentiate the denominator $v=n-1$ to get $v'=1$ .
Apply quotient rule: Apply the quotient rule: $a'_{n}=\frac{(n-1)(2n-2)-(n^2-2n+1)\cdot 1}{(n-1)^2}.$
Simplify derivative expression: Simplify the derivative expression: $a'_{n}=\frac{2n^2-2n-2n+2-n^2+2n-1}{(n-1)^2}$ .
Combine like terms: Combine like terms in the numerator: $a'_{n}=\frac{n^2-1}{(n-1)^2}$ .
Recognize difference of squares: Realize that the numerator is a difference of squares: $a'_{n}=\frac{(n-1)(n+1)}{(n-1)^2}.$
Cancel common term: Cancel out the common $(n-1)$ term in the numerator and denominator: $a'_{n}=\frac{n+1}{n-1}$ .

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