Solve 2r/r^2-1-1/r+1

$$\frac { 2 r } { r ^ { 2 } - 1 } - \frac { 1 } { r + 1 }$$

Evaluate

$$\frac{1}{r-1}$$

Short Solution Steps

Factor r^{2}-1.

$$\frac{2r}{\left(r-1\right)\left(r+1\right)}-\frac{1}{r+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(r-1\right)\left(r+1\right) and r+1 is \left(r-1\right)\left(r+1\right). Multiply \frac{1}{r+1} times \frac{r-1}{r-1}.

$$\frac{2r}{\left(r-1\right)\left(r+1\right)}-\frac{r-1}{\left(r-1\right)\left(r+1\right)}$$

Since \frac{2r}{\left(r-1\right)\left(r+1\right)} and \frac{r-1}{\left(r-1\right)\left(r+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2r-\left(r-1\right)}{\left(r-1\right)\left(r+1\right)}$$

Do the multiplications in 2r-\left(r-1\right).

$$\frac{2r-r+1}{\left(r-1\right)\left(r+1\right)}$$

Combine like terms in 2r-r+1.

$$\frac{r+1}{\left(r-1\right)\left(r+1\right)}$$

Cancel out r+1 in both numerator and denominator.

$$\frac{1}{r-1}$$

Differentiate w.r.t. r

$$-\frac{1}{\left(r-1\right)^{2}}$$

Steps Using Chain Rule

Factor r^{2}-1.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r}{\left(r-1\right)\left(r+1\right)}-\frac{1}{r+1})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(r-1\right)\left(r+1\right) and r+1 is \left(r-1\right)\left(r+1\right). Multiply \frac{1}{r+1} times \frac{r-1}{r-1}.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r}{\left(r-1\right)\left(r+1\right)}-\frac{r-1}{\left(r-1\right)\left(r+1\right)})$$

Since \frac{2r}{\left(r-1\right)\left(r+1\right)} and \frac{r-1}{\left(r-1\right)\left(r+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r-\left(r-1\right)}{\left(r-1\right)\left(r+1\right)})$$

Do the multiplications in 2r-\left(r-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2r-r+1}{\left(r-1\right)\left(r+1\right)})$$

Combine like terms in 2r-r+1.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{r+1}{\left(r-1\right)\left(r+1\right)})$$

Cancel out r+1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{r-1})$$

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).

$$-\left(r^{1}-1\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}r}(r^{1}-1)$$

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}.

$$-\left(r^{1}-1\right)^{-2}r^{1-1}$$

Simplify.

$$-r^{0}\left(r^{1}-1\right)^{-2}$$

For any term t, t^{1}=t.

$$-r^{0}\left(r-1\right)^{-2}$$

For any term t except 0, t^{0}=1.

$$-\left(r-1\right)^{-2}$$