Solve 12/w^2-22w-23-11/w^2-24w+23

$$\frac { 12 } { w ^ { 2 } - 22 w - 23 } - \frac { 11 } { w ^ { 2 } - 24 w + 23 }$$

Evaluate

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

Short Solution Steps

Factor w^{2}-22w-23. Factor w^{2}-24w+23.

$$\frac{12}{\left(w-23\right)\left(w+1\right)}-\frac{11}{\left(w-23\right)\left(w-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(w-23\right)\left(w+1\right) and \left(w-23\right)\left(w-1\right) is \left(w-23\right)\left(w-1\right)\left(w+1\right). Multiply \frac{12}{\left(w-23\right)\left(w+1\right)} times \frac{w-1}{w-1}. Multiply \frac{11}{\left(w-23\right)\left(w-1\right)} times \frac{w+1}{w+1}.

$$\frac{12\left(w-1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)}-\frac{11\left(w+1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)}$$

Since \frac{12\left(w-1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)} and \frac{11\left(w+1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{12\left(w-1\right)-11\left(w+1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)}$$

Do the multiplications in 12\left(w-1\right)-11\left(w+1\right).

$$\frac{12w-12-11w-11}{\left(w-23\right)\left(w-1\right)\left(w+1\right)}$$

Combine like terms in 12w-12-11w-11.

$$\frac{w-23}{\left(w-23\right)\left(w-1\right)\left(w+1\right)}$$

Cancel out w-23 in both numerator and denominator.

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

Expand \left(w-1\right)\left(w+1\right).

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

Differentiate w.r.t. w

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

Steps Using Chain Rule

Factor w^{2}-22w-23. Factor w^{2}-24w+23.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{12}{\left(w-23\right)\left(w+1\right)}-\frac{11}{\left(w-23\right)\left(w-1\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(w-23\right)\left(w+1\right) and \left(w-23\right)\left(w-1\right) is \left(w-23\right)\left(w-1\right)\left(w+1\right). Multiply \frac{12}{\left(w-23\right)\left(w+1\right)} times \frac{w-1}{w-1}. Multiply \frac{11}{\left(w-23\right)\left(w-1\right)} times \frac{w+1}{w+1}.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{12\left(w-1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)}-\frac{11\left(w+1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)})$$

Since \frac{12\left(w-1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)} and \frac{11\left(w+1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{12\left(w-1\right)-11\left(w+1\right)}{\left(w-23\right)\left(w-1\right)\left(w+1\right)})$$

Do the multiplications in 12\left(w-1\right)-11\left(w+1\right).

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{12w-12-11w-11}{\left(w-23\right)\left(w-1\right)\left(w+1\right)})$$

Combine like terms in 12w-12-11w-11.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{w-23}{\left(w-23\right)\left(w-1\right)\left(w+1\right)})$$

Cancel out w-23 in both numerator and denominator.

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

Consider \left(w-1\right)\left(w+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{1}{w^{2}-1})$$

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).

$$-\left(w^{2}-1\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}w}(w^{2}-1)$$

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}.

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

Simplify.

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

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

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