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Differentiate w.r.t. m
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\frac{2mn}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}+\frac{2m}{\left(m+n\right)\left(m-n\right)}-\frac{1}{m-n}
Factor m^{3}+n^{3}. Factor m^{2}-n^{2}.
\frac{2mn\left(m-n\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}+\frac{2m\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m+n\right)\left(m^{2}-mn+n^{2}\right) and \left(m+n\right)\left(m-n\right) is \left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right). Multiply \frac{2mn}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)} times \frac{m-n}{m-n}. Multiply \frac{2m}{\left(m+n\right)\left(m-n\right)} times \frac{m^{2}-mn+n^{2}}{m^{2}-mn+n^{2}}.
\frac{2mn\left(m-n\right)+2m\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}
Since \frac{2mn\left(m-n\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} and \frac{2m\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} have the same denominator, add them by adding their numerators.
\frac{2m^{2}n-2mn^{2}+2m^{3}-2m^{2}n+2mn^{2}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}
Do the multiplications in 2mn\left(m-n\right)+2m\left(m^{2}-mn+n^{2}\right).
\frac{2m^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}
Combine like terms in 2m^{2}n-2mn^{2}+2m^{3}-2m^{2}n+2mn^{2}.
\frac{2m^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right) and m-n is \left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right). Multiply \frac{1}{m-n} times \frac{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}.
\frac{2m^{3}-\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}
Since \frac{2m^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} and \frac{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2m^{3}-m^{3}+m^{2}n-mn^{2}-nm^{2}+n^{2}m-n^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}
Do the multiplications in 2m^{3}-\left(m+n\right)\left(m^{2}-mn+n^{2}\right).
\frac{m^{3}-n^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}
Combine like terms in 2m^{3}-m^{3}+m^{2}n-mn^{2}-nm^{2}+n^{2}m-n^{3}.
\frac{\left(m-n\right)\left(m^{2}+mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}
Factor the expressions that are not already factored in \frac{m^{3}-n^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}.
\frac{m^{2}+mn+n^{2}}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}
Cancel out m-n in both numerator and denominator.
\frac{m^{2}+mn+n^{2}}{m^{3}+n^{3}}
Expand \left(m+n\right)\left(m^{2}-mn+n^{2}\right).