Solve m/m^2-1-1/1-m^2

$$\frac { m } { m ^ { 2 } - 1 } - \frac { 1 } { 1 - m ^ { 2 } }$$

Evaluate

$$-\frac{1}{1-m}$$

Short Solution Steps

Factor m^{2}-1. Factor 1-m^{2}.

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

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

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

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

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

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

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

Cancel out m+1 in both numerator and denominator.

$$\frac{1}{m-1}$$

Differentiate w.r.t. m

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

Steps Using Chain Rule

Factor m^{2}-1. Factor 1-m^{2}.

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

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

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

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

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

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

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

Cancel out m+1 in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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

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

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

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

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