Solve 1/1,3^2-x-2/1,3^2-1+1/1,3^2+x

$$\frac { 1 } { 1,3 ^ { 2 } - x } - \frac { 2 } { 1,3 ^ { 2 } - 1 } + \frac { 1 } { 1,3 ^ { 2 } + x }$$

Evaluate

$$\frac{x^{2}-9}{4\left(81-x^{2}\right)}$$

Short Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{1\times 9-x}-\frac{2}{1\times 3^{2}-1}+\frac{1}{1\times 3^{2}+x}$$

Multiply 1 and 9 to get 9.

$$\frac{1}{9-x}-\frac{2}{1\times 3^{2}-1}+\frac{1}{1\times 3^{2}+x}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{9-x}-\frac{2}{1\times 9-1}+\frac{1}{1\times 3^{2}+x}$$

Multiply 1 and 9 to get 9.

$$\frac{1}{9-x}-\frac{2}{9-1}+\frac{1}{1\times 3^{2}+x}$$

Subtract 1 from 9 to get 8.

$$\frac{1}{9-x}-\frac{2}{8}+\frac{1}{1\times 3^{2}+x}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{1}{9-x}-\frac{1}{4}+\frac{1}{1\times 3^{2}+x}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{9-x}-\frac{1}{4}+\frac{1}{1\times 9+x}$$

Multiply 1 and 9 to get 9.

$$\frac{1}{9-x}-\frac{1}{4}+\frac{1}{9+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9-x and 4 is 4\left(-x+9\right). Multiply \frac{1}{9-x} times \frac{4}{4}. Multiply \frac{1}{4} times \frac{-x+9}{-x+9}.

$$\frac{4}{4\left(-x+9\right)}-\frac{-x+9}{4\left(-x+9\right)}+\frac{1}{9+x}$$

Since \frac{4}{4\left(-x+9\right)} and \frac{-x+9}{4\left(-x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-\left(-x+9\right)}{4\left(-x+9\right)}+\frac{1}{9+x}$$

Do the multiplications in 4-\left(-x+9\right).

$$\frac{4+x-9}{4\left(-x+9\right)}+\frac{1}{9+x}$$

Combine like terms in 4+x-9.

$$\frac{-5+x}{4\left(-x+9\right)}+\frac{1}{9+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\left(-x+9\right) and 9+x is 4\left(x+9\right)\left(-x+9\right). Multiply \frac{-5+x}{4\left(-x+9\right)} times \frac{x+9}{x+9}. Multiply \frac{1}{9+x} times \frac{4\left(-x+9\right)}{4\left(-x+9\right)}.

$$\frac{\left(-5+x\right)\left(x+9\right)}{4\left(x+9\right)\left(-x+9\right)}+\frac{4\left(-x+9\right)}{4\left(x+9\right)\left(-x+9\right)}$$

Since \frac{\left(-5+x\right)\left(x+9\right)}{4\left(x+9\right)\left(-x+9\right)} and \frac{4\left(-x+9\right)}{4\left(x+9\right)\left(-x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-5+x\right)\left(x+9\right)+4\left(-x+9\right)}{4\left(x+9\right)\left(-x+9\right)}$$

Do the multiplications in \left(-5+x\right)\left(x+9\right)+4\left(-x+9\right).

$$\frac{-5x-45+x^{2}+9x-4x+36}{4\left(x+9\right)\left(-x+9\right)}$$

Combine like terms in -5x-45+x^{2}+9x-4x+36.

$$\frac{-9+x^{2}}{4\left(x+9\right)\left(-x+9\right)}$$

Expand 4\left(x+9\right)\left(-x+9\right).

$$\frac{-9+x^{2}}{-4x^{2}+324}$$

Differentiate w.r.t. x

$$\frac{36x}{\left(x^{2}-81\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{1\times 9-x}-\frac{2}{1\times 3^{2}-1}+\frac{1}{1\times 3^{2}+x})$$

Multiply 1 and 9 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{2}{1\times 3^{2}-1}+\frac{1}{1\times 3^{2}+x})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{2}{1\times 9-1}+\frac{1}{1\times 3^{2}+x})$$

Multiply 1 and 9 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{2}{9-1}+\frac{1}{1\times 3^{2}+x})$$

Subtract 1 from 9 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{2}{8}+\frac{1}{1\times 3^{2}+x})$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{1}{4}+\frac{1}{1\times 3^{2}+x})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{1}{4}+\frac{1}{1\times 9+x})$$

Multiply 1 and 9 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9-x}-\frac{1}{4}+\frac{1}{9+x})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9-x and 4 is 4\left(-x+9\right). Multiply \frac{1}{9-x} times \frac{4}{4}. Multiply \frac{1}{4} times \frac{-x+9}{-x+9}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{4\left(-x+9\right)}-\frac{-x+9}{4\left(-x+9\right)}+\frac{1}{9+x})$$

Since \frac{4}{4\left(-x+9\right)} and \frac{-x+9}{4\left(-x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4-\left(-x+9\right)}{4\left(-x+9\right)}+\frac{1}{9+x})$$

Do the multiplications in 4-\left(-x+9\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4+x-9}{4\left(-x+9\right)}+\frac{1}{9+x})$$

Combine like terms in 4+x-9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-5+x}{4\left(-x+9\right)}+\frac{1}{9+x})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\left(-x+9\right) and 9+x is 4\left(x+9\right)\left(-x+9\right). Multiply \frac{-5+x}{4\left(-x+9\right)} times \frac{x+9}{x+9}. Multiply \frac{1}{9+x} times \frac{4\left(-x+9\right)}{4\left(-x+9\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(-5+x\right)\left(x+9\right)}{4\left(x+9\right)\left(-x+9\right)}+\frac{4\left(-x+9\right)}{4\left(x+9\right)\left(-x+9\right)})$$

Since \frac{\left(-5+x\right)\left(x+9\right)}{4\left(x+9\right)\left(-x+9\right)} and \frac{4\left(-x+9\right)}{4\left(x+9\right)\left(-x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(-5+x\right)\left(x+9\right)+4\left(-x+9\right)}{4\left(x+9\right)\left(-x+9\right)})$$

Do the multiplications in \left(-5+x\right)\left(x+9\right)+4\left(-x+9\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-5x-45+x^{2}+9x-4x+36}{4\left(x+9\right)\left(-x+9\right)})$$

Combine like terms in -5x-45+x^{2}+9x-4x+36.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-9+x^{2}}{4\left(x+9\right)\left(-x+9\right)})$$

Use the distributive property to multiply 4 by x+9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-9+x^{2}}{\left(4x+36\right)\left(-x+9\right)})$$

Apply the distributive property by multiplying each term of 4x+36 by each term of -x+9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-9+x^{2}}{-4x^{2}+36x-36x+324})$$

Combine 36x and -36x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-9+x^{2}}{-4x^{2}+324})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(-4x^{2}+324\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-9)-\left(x^{2}-9\right)\frac{\mathrm{d}}{\mathrm{d}x}(-4x^{2}+324)}{\left(-4x^{2}+324\right)^{2}}$$

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}.

$$\frac{\left(-4x^{2}+324\right)\times 2x^{2-1}-\left(x^{2}-9\right)\times 2\left(-4\right)x^{2-1}}{\left(-4x^{2}+324\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(-4x^{2}+324\right)\times 2x^{1}-\left(x^{2}-9\right)\left(-8\right)x^{1}}{\left(-4x^{2}+324\right)^{2}}$$

Expand using distributive property.

$$\frac{-4x^{2}\times 2x^{1}+324\times 2x^{1}-\left(x^{2}\left(-8\right)x^{1}-9\left(-8\right)x^{1}\right)}{\left(-4x^{2}+324\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{-4\times 2x^{2+1}+324\times 2x^{1}-\left(-8x^{2+1}-9\left(-8\right)x^{1}\right)}{\left(-4x^{2}+324\right)^{2}}$$

Do the arithmetic.

$$\frac{-8x^{3}+648x^{1}-\left(-8x^{3}+72x^{1}\right)}{\left(-4x^{2}+324\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{-8x^{3}+648x^{1}-\left(-8x^{3}\right)-72x^{1}}{\left(-4x^{2}+324\right)^{2}}$$

Combine like terms.

$$\frac{\left(-8-\left(-8\right)\right)x^{3}+\left(648-72\right)x^{1}}{\left(-4x^{2}+324\right)^{2}}$$

Subtract -8 from -8 and 72 from 648.

$$\frac{576x^{1}}{\left(-4x^{2}+324\right)^{2}}$$

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

$$\frac{576x}{\left(-4x^{2}+324\right)^{2}}$$