Solve f(z)=7+2/z+6-6/z^2-36

$$f ( z ) = 7 + \frac { 2 } { z + 6 } - \frac { 6 } { z ^ { 2 } - 36 }$$

Evaluate

$$\frac{7z^{2}+2z-270}{z^{2}-36}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{z+6}{z+6}.

$$\frac{7\left(z+6\right)}{z+6}+\frac{2}{z+6}-\frac{6}{z^{2}-36}$$

Since \frac{7\left(z+6\right)}{z+6} and \frac{2}{z+6} have the same denominator, add them by adding their numerators.

$$\frac{7\left(z+6\right)+2}{z+6}-\frac{6}{z^{2}-36}$$

Do the multiplications in 7\left(z+6\right)+2.

$$\frac{7z+42+2}{z+6}-\frac{6}{z^{2}-36}$$

Combine like terms in 7z+42+2.

$$\frac{7z+44}{z+6}-\frac{6}{z^{2}-36}$$

Factor z^{2}-36.

$$\frac{7z+44}{z+6}-\frac{6}{\left(z-6\right)\left(z+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z+6 and \left(z-6\right)\left(z+6\right) is \left(z-6\right)\left(z+6\right). Multiply \frac{7z+44}{z+6} times \frac{z-6}{z-6}.

$$\frac{\left(7z+44\right)\left(z-6\right)}{\left(z-6\right)\left(z+6\right)}-\frac{6}{\left(z-6\right)\left(z+6\right)}$$

Since \frac{\left(7z+44\right)\left(z-6\right)}{\left(z-6\right)\left(z+6\right)} and \frac{6}{\left(z-6\right)\left(z+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(7z+44\right)\left(z-6\right)-6}{\left(z-6\right)\left(z+6\right)}$$

Do the multiplications in \left(7z+44\right)\left(z-6\right)-6.

$$\frac{7z^{2}-42z+44z-264-6}{\left(z-6\right)\left(z+6\right)}$$

Combine like terms in 7z^{2}-42z+44z-264-6.

$$\frac{7z^{2}+2z-270}{\left(z-6\right)\left(z+6\right)}$$

Expand \left(z-6\right)\left(z+6\right).

$$\frac{7z^{2}+2z-270}{z^{2}-36}$$

Differentiate w.r.t. z

$$\frac{2\left(-z^{2}+18z-36\right)}{\left(z^{2}-36\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{z+6}{z+6}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7\left(z+6\right)}{z+6}+\frac{2}{z+6}-\frac{6}{z^{2}-36})$$

Since \frac{7\left(z+6\right)}{z+6} and \frac{2}{z+6} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7\left(z+6\right)+2}{z+6}-\frac{6}{z^{2}-36})$$

Do the multiplications in 7\left(z+6\right)+2.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7z+42+2}{z+6}-\frac{6}{z^{2}-36})$$

Combine like terms in 7z+42+2.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7z+44}{z+6}-\frac{6}{z^{2}-36})$$

Factor z^{2}-36.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7z+44}{z+6}-\frac{6}{\left(z-6\right)\left(z+6\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z+6 and \left(z-6\right)\left(z+6\right) is \left(z-6\right)\left(z+6\right). Multiply \frac{7z+44}{z+6} times \frac{z-6}{z-6}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\left(7z+44\right)\left(z-6\right)}{\left(z-6\right)\left(z+6\right)}-\frac{6}{\left(z-6\right)\left(z+6\right)})$$

Since \frac{\left(7z+44\right)\left(z-6\right)}{\left(z-6\right)\left(z+6\right)} and \frac{6}{\left(z-6\right)\left(z+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\left(7z+44\right)\left(z-6\right)-6}{\left(z-6\right)\left(z+6\right)})$$

Do the multiplications in \left(7z+44\right)\left(z-6\right)-6.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7z^{2}-42z+44z-264-6}{\left(z-6\right)\left(z+6\right)})$$

Combine like terms in 7z^{2}-42z+44z-264-6.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7z^{2}+2z-270}{\left(z-6\right)\left(z+6\right)})$$

Consider \left(z-6\right)\left(z+6\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 6.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{7z^{2}+2z-270}{z^{2}-36})$$

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.

$$\frac{\left(z^{2}-36\right)\frac{\mathrm{d}}{\mathrm{d}z}(7z^{2}+2z^{1}-270)-\left(7z^{2}+2z^{1}-270\right)\frac{\mathrm{d}}{\mathrm{d}z}(z^{2}-36)}{\left(z^{2}-36\right)^{2}}$$

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}.

$$\frac{\left(z^{2}-36\right)\left(2\times 7z^{2-1}+2z^{1-1}\right)-\left(7z^{2}+2z^{1}-270\right)\times 2z^{2-1}}{\left(z^{2}-36\right)^{2}}$$

Simplify.

$$\frac{\left(z^{2}-36\right)\left(14z^{1}+2z^{0}\right)-\left(7z^{2}+2z^{1}-270\right)\times 2z^{1}}{\left(z^{2}-36\right)^{2}}$$

Multiply z^{2}-36 times 14z^{1}+2z^{0}.

$$\frac{z^{2}\times 14z^{1}+z^{2}\times 2z^{0}-36\times 14z^{1}-36\times 2z^{0}-\left(7z^{2}+2z^{1}-270\right)\times 2z^{1}}{\left(z^{2}-36\right)^{2}}$$

Multiply 7z^{2}+2z^{1}-270 times 2z^{1}.

$$\frac{z^{2}\times 14z^{1}+z^{2}\times 2z^{0}-36\times 14z^{1}-36\times 2z^{0}-\left(7z^{2}\times 2z^{1}+2z^{1}\times 2z^{1}-270\times 2z^{1}\right)}{\left(z^{2}-36\right)^{2}}$$

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

$$\frac{14z^{2+1}+2z^{2}-36\times 14z^{1}-36\times 2z^{0}-\left(7\times 2z^{2+1}+2\times 2z^{1+1}-270\times 2z^{1}\right)}{\left(z^{2}-36\right)^{2}}$$

Simplify.

$$\frac{14z^{3}+2z^{2}-504z^{1}-72z^{0}-\left(14z^{3}+4z^{2}-540z^{1}\right)}{\left(z^{2}-36\right)^{2}}$$

Combine like terms.

$$\frac{-2z^{2}+36z^{1}-72z^{0}}{\left(z^{2}-36\right)^{2}}$$

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

$$\frac{-2z^{2}+36z-72z^{0}}{\left(z^{2}-36\right)^{2}}$$

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

$$\frac{-2z^{2}+36z-72}{\left(z^{2}-36\right)^{2}}$$