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Differentiate w.r.t. x
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\frac{a\left(-2x+a\right)}{\left(-2x+a\right)\left(2x+a\right)}+\frac{2x\left(2x+a\right)}{\left(-2x+a\right)\left(2x+a\right)}+\frac{4ax}{a^{2}-4x^{2}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+2x and a-2x is \left(-2x+a\right)\left(2x+a\right). Multiply \frac{a}{a+2x} times \frac{-2x+a}{-2x+a}. Multiply \frac{2x}{a-2x} times \frac{2x+a}{2x+a}.
\frac{a\left(-2x+a\right)+2x\left(2x+a\right)}{\left(-2x+a\right)\left(2x+a\right)}+\frac{4ax}{a^{2}-4x^{2}}
Since \frac{a\left(-2x+a\right)}{\left(-2x+a\right)\left(2x+a\right)} and \frac{2x\left(2x+a\right)}{\left(-2x+a\right)\left(2x+a\right)} have the same denominator, add them by adding their numerators.
\frac{-2ax+a^{2}+4x^{2}+2xa}{\left(-2x+a\right)\left(2x+a\right)}+\frac{4ax}{a^{2}-4x^{2}}
Do the multiplications in a\left(-2x+a\right)+2x\left(2x+a\right).
\frac{4x^{2}+a^{2}}{\left(-2x+a\right)\left(2x+a\right)}+\frac{4ax}{a^{2}-4x^{2}}
Combine like terms in -2ax+a^{2}+4x^{2}+2xa.
\frac{4x^{2}+a^{2}}{\left(-2x+a\right)\left(2x+a\right)}+\frac{4ax}{\left(-2x+a\right)\left(2x+a\right)}
Factor a^{2}-4x^{2}.
\frac{4x^{2}+a^{2}+4ax}{\left(-2x+a\right)\left(2x+a\right)}
Since \frac{4x^{2}+a^{2}}{\left(-2x+a\right)\left(2x+a\right)} and \frac{4ax}{\left(-2x+a\right)\left(2x+a\right)} have the same denominator, add them by adding their numerators.
\frac{\left(2x+a\right)^{2}}{\left(-2x+a\right)\left(2x+a\right)}
Factor the expressions that are not already factored in \frac{4x^{2}+a^{2}+4ax}{\left(-2x+a\right)\left(2x+a\right)}.
\frac{2x+a}{-2x+a}
Cancel out 2x+a in both numerator and denominator.