To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $q+1$ and $\left(q-1\right)\left(q+1\right)$ is $\left(q-1\right)\left(q+1\right)$. Multiply $\frac{1}{q+1}$ times $\frac{q-1}{q-1}$.
Since $\frac{q-1}{\left(q-1\right)\left(q+1\right)}$ and $\frac{q}{\left(q-1\right)\left(q+1\right)}$ have the same denominator, subtract them by subtracting their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(q-1\right)\left(q+1\right)$ and $\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)$ is $\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)$. Multiply $\frac{-1}{\left(q-1\right)\left(q+1\right)}$ times $\frac{-q^{2}-1}{-q^{2}-1}$.
Since $\frac{-\left(-q^{2}-1\right)}{\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)}$ and $\frac{q^{2}}{\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)}$ have the same denominator, subtract them by subtracting their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $q+1$ and $\left(q-1\right)\left(q+1\right)$ is $\left(q-1\right)\left(q+1\right)$. Multiply $\frac{1}{q+1}$ times $\frac{q-1}{q-1}$.
Since $\frac{q-1}{\left(q-1\right)\left(q+1\right)}$ and $\frac{q}{\left(q-1\right)\left(q+1\right)}$ have the same denominator, subtract them by subtracting their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(q-1\right)\left(q+1\right)$ and $\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)$ is $\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)$. Multiply $\frac{-1}{\left(q-1\right)\left(q+1\right)}$ times $\frac{-q^{2}-1}{-q^{2}-1}$.
Since $\frac{-\left(-q^{2}-1\right)}{\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)}$ and $\frac{q^{2}}{\left(q-1\right)\left(q+1\right)\left(-q^{2}-1\right)}$ have the same denominator, subtract them by subtracting their numerators.
If $F$ is the composition of two differentiable functions $f\left(u\right)$ and $u=g\left(x\right)$, that is, if $F\left(x\right)=f\left(g\left(x\right)\right)$, then the derivative of $F$ is the derivative of $f$ with respect to $u$ times the derivative of $g$ with respect to $x$, that is, $\frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right)$.
The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is $0$. The derivative of $ax^{n}$ is $nax^{n-1}$.
