Solve 8/m-9+1/m^2-81

$$\frac { 8 } { m - 9 } + \frac { 1 } { m ^ { 2 } - 81 }$$

Evaluate

$$\frac{8m+73}{m^{2}-81}$$

Short Solution Steps

Factor m^{2}-81.

$$\frac{8}{m-9}+\frac{1}{\left(m-9\right)\left(m+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m-9 and \left(m-9\right)\left(m+9\right) is \left(m-9\right)\left(m+9\right). Multiply \frac{8}{m-9} times \frac{m+9}{m+9}.

$$\frac{8\left(m+9\right)}{\left(m-9\right)\left(m+9\right)}+\frac{1}{\left(m-9\right)\left(m+9\right)}$$

Since \frac{8\left(m+9\right)}{\left(m-9\right)\left(m+9\right)} and \frac{1}{\left(m-9\right)\left(m+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(m+9\right)+1}{\left(m-9\right)\left(m+9\right)}$$

Do the multiplications in 8\left(m+9\right)+1.

$$\frac{8m+72+1}{\left(m-9\right)\left(m+9\right)}$$

Combine like terms in 8m+72+1.

$$\frac{8m+73}{\left(m-9\right)\left(m+9\right)}$$

Expand \left(m-9\right)\left(m+9\right).

$$\frac{8m+73}{m^{2}-81}$$

Differentiate w.r.t. m

$$-\frac{2\left(4m^{2}+73m+324\right)}{\left(m^{2}-81\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor m^{2}-81.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8}{m-9}+\frac{1}{\left(m-9\right)\left(m+9\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m-9 and \left(m-9\right)\left(m+9\right) is \left(m-9\right)\left(m+9\right). Multiply \frac{8}{m-9} times \frac{m+9}{m+9}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8\left(m+9\right)}{\left(m-9\right)\left(m+9\right)}+\frac{1}{\left(m-9\right)\left(m+9\right)})$$

Since \frac{8\left(m+9\right)}{\left(m-9\right)\left(m+9\right)} and \frac{1}{\left(m-9\right)\left(m+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8\left(m+9\right)+1}{\left(m-9\right)\left(m+9\right)})$$

Do the multiplications in 8\left(m+9\right)+1.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8m+72+1}{\left(m-9\right)\left(m+9\right)})$$

Combine like terms in 8m+72+1.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8m+73}{\left(m-9\right)\left(m+9\right)})$$

Consider \left(m-9\right)\left(m+9\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 9.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8m+73}{m^{2}-81})$$

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.

$$\frac{\left(m^{2}-81\right)\frac{\mathrm{d}}{\mathrm{d}m}(8m^{1}+73)-\left(8m^{1}+73\right)\frac{\mathrm{d}}{\mathrm{d}m}(m^{2}-81)}{\left(m^{2}-81\right)^{2}}$$

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}.

$$\frac{\left(m^{2}-81\right)\times 8m^{1-1}-\left(8m^{1}+73\right)\times 2m^{2-1}}{\left(m^{2}-81\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(m^{2}-81\right)\times 8m^{0}-\left(8m^{1}+73\right)\times 2m^{1}}{\left(m^{2}-81\right)^{2}}$$

Expand using distributive property.

$$\frac{m^{2}\times 8m^{0}-81\times 8m^{0}-\left(8m^{1}\times 2m^{1}+73\times 2m^{1}\right)}{\left(m^{2}-81\right)^{2}}$$

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

$$\frac{8m^{2}-81\times 8m^{0}-\left(8\times 2m^{1+1}+73\times 2m^{1}\right)}{\left(m^{2}-81\right)^{2}}$$

Do the arithmetic.

$$\frac{8m^{2}-648m^{0}-\left(16m^{2}+146m^{1}\right)}{\left(m^{2}-81\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{8m^{2}-648m^{0}-16m^{2}-146m^{1}}{\left(m^{2}-81\right)^{2}}$$

Combine like terms.

$$\frac{\left(8-16\right)m^{2}-648m^{0}-146m^{1}}{\left(m^{2}-81\right)^{2}}$$

Subtract 16 from 8.

$$\frac{-8m^{2}-648m^{0}-146m^{1}}{\left(m^{2}-81\right)^{2}}$$

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

$$\frac{-8m^{2}-648m^{0}-146m}{\left(m^{2}-81\right)^{2}}$$

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

$$\frac{-8m^{2}-648-146m}{\left(m^{2}-81\right)^{2}}$$