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Differentiate w.r.t. k
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\frac{k}{k+1}+\frac{1}{\frac{\left(k+2\right)k}{k}+\frac{1}{k}}
To add or subtract expressions, expand them to make their denominators the same. Multiply k+2 times \frac{k}{k}.
\frac{k}{k+1}+\frac{1}{\frac{\left(k+2\right)k+1}{k}}
Since \frac{\left(k+2\right)k}{k} and \frac{1}{k} have the same denominator, add them by adding their numerators.
\frac{k}{k+1}+\frac{1}{\frac{k^{2}+2k+1}{k}}
Do the multiplications in \left(k+2\right)k+1.
\frac{k}{k+1}+\frac{k}{k^{2}+2k+1}
Divide 1 by \frac{k^{2}+2k+1}{k} by multiplying 1 by the reciprocal of \frac{k^{2}+2k+1}{k}.
\frac{k}{k+1}+\frac{k}{\left(k+1\right)^{2}}
Factor k^{2}+2k+1.
\frac{k\left(k+1\right)}{\left(k+1\right)^{2}}+\frac{k}{\left(k+1\right)^{2}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of k+1 and \left(k+1\right)^{2} is \left(k+1\right)^{2}. Multiply \frac{k}{k+1} times \frac{k+1}{k+1}.
\frac{k\left(k+1\right)+k}{\left(k+1\right)^{2}}
Since \frac{k\left(k+1\right)}{\left(k+1\right)^{2}} and \frac{k}{\left(k+1\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{k^{2}+k+k}{\left(k+1\right)^{2}}
Do the multiplications in k\left(k+1\right)+k.
\frac{k^{2}+2k}{\left(k+1\right)^{2}}
Combine like terms in k^{2}+k+k.
\frac{k^{2}+2k}{k^{2}+2k+1}
Expand \left(k+1\right)^{2}.