Solve 15/b^2-4*2/b+2/frac{17{b^2-4}*15/b-2}

$$\frac { \frac { 15 } { b ^ { 2 } - 4 } \cdot \frac { 2 } { b + 2 } } { \frac { 17 } { b ^ { 2 } - 4 } \cdot \frac { 15 } { b - 2 } }$$

Evaluate

$$\frac{2\left(b-2\right)}{17\left(b+2\right)}$$

Short Solution Steps

Multiply \frac{15}{b^{2}-4} times \frac{2}{b+2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}\times \frac{15}{b-2}}$$

Multiply \frac{17}{b^{2}-4} times \frac{15}{b-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)}}{\frac{17\times 15}{\left(b^{2}-4\right)\left(b-2\right)}}$$

Divide \frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)} by \frac{17\times 15}{\left(b^{2}-4\right)\left(b-2\right)} by multiplying \frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)} by the reciprocal of \frac{17\times 15}{\left(b^{2}-4\right)\left(b-2\right)}.

$$\frac{15\times 2\left(b^{2}-4\right)\left(b-2\right)}{\left(b^{2}-4\right)\left(b+2\right)\times 17\times 15}$$

Cancel out 15\left(b^{2}-4\right) in both numerator and denominator.

$$\frac{2\left(b-2\right)}{17\left(b+2\right)}$$

Use the distributive property to multiply 2 by b-2.

$$\frac{2b-4}{17\left(b+2\right)}$$

Use the distributive property to multiply 17 by b+2.

$$\frac{2b-4}{17b+34}$$

Differentiate w.r.t. b

$$\frac{8}{17\left(b+2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Multiply \frac{15}{b^{2}-4} times \frac{2}{b+2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)}}{\frac{17}{b^{2}-4}\times \frac{15}{b-2}})$$

Multiply \frac{17}{b^{2}-4} times \frac{15}{b-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)}}{\frac{17\times 15}{\left(b^{2}-4\right)\left(b-2\right)}})$$

Divide \frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)} by \frac{17\times 15}{\left(b^{2}-4\right)\left(b-2\right)} by multiplying \frac{15\times 2}{\left(b^{2}-4\right)\left(b+2\right)} by the reciprocal of \frac{17\times 15}{\left(b^{2}-4\right)\left(b-2\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{15\times 2\left(b^{2}-4\right)\left(b-2\right)}{\left(b^{2}-4\right)\left(b+2\right)\times 17\times 15})$$

Cancel out 15\left(b^{2}-4\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{2\left(b-2\right)}{17\left(b+2\right)})$$

Use the distributive property to multiply 2 by b-2.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{2b-4}{17\left(b+2\right)})$$

Use the distributive property to multiply 17 by b+2.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{2b-4}{17b+34})$$

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.

$$\frac{\left(17b^{1}+34\right)\frac{\mathrm{d}}{\mathrm{d}b}(2b^{1}-4)-\left(2b^{1}-4\right)\frac{\mathrm{d}}{\mathrm{d}b}(17b^{1}+34)}{\left(17b^{1}+34\right)^{2}}$$

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}.

$$\frac{\left(17b^{1}+34\right)\times 2b^{1-1}-\left(2b^{1}-4\right)\times 17b^{1-1}}{\left(17b^{1}+34\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(17b^{1}+34\right)\times 2b^{0}-\left(2b^{1}-4\right)\times 17b^{0}}{\left(17b^{1}+34\right)^{2}}$$

Expand using distributive property.

$$\frac{17b^{1}\times 2b^{0}+34\times 2b^{0}-\left(2b^{1}\times 17b^{0}-4\times 17b^{0}\right)}{\left(17b^{1}+34\right)^{2}}$$

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

$$\frac{17\times 2b^{1}+34\times 2b^{0}-\left(2\times 17b^{1}-4\times 17b^{0}\right)}{\left(17b^{1}+34\right)^{2}}$$

Do the arithmetic.

$$\frac{34b^{1}+68b^{0}-\left(34b^{1}-68b^{0}\right)}{\left(17b^{1}+34\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{34b^{1}+68b^{0}-34b^{1}-\left(-68b^{0}\right)}{\left(17b^{1}+34\right)^{2}}$$

Combine like terms.

$$\frac{\left(34-34\right)b^{1}+\left(68-\left(-68\right)\right)b^{0}}{\left(17b^{1}+34\right)^{2}}$$

Subtract 34 from 34 and -68 from 68.

$$\frac{136b^{0}}{\left(17b^{1}+34\right)^{2}}$$

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

$$\frac{136b^{0}}{\left(17b+34\right)^{2}}$$

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

$$\frac{136\times 1}{\left(17b+34\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{136}{\left(17b+34\right)^{2}}$$