Solve w/3w-7+4/3w+7

$$\frac { w } { 3 w - 7 } + \frac { 4 } { 3 w + 7 }$$

Evaluate

$$\frac{3w^{2}+19w-28}{9w^{2}-49}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3w-7 and 3w+7 is \left(3w-7\right)\left(3w+7\right). Multiply \frac{w}{3w-7} times \frac{3w+7}{3w+7}. Multiply \frac{4}{3w+7} times \frac{3w-7}{3w-7}.

$$\frac{w\left(3w+7\right)}{\left(3w-7\right)\left(3w+7\right)}+\frac{4\left(3w-7\right)}{\left(3w-7\right)\left(3w+7\right)}$$

Since \frac{w\left(3w+7\right)}{\left(3w-7\right)\left(3w+7\right)} and \frac{4\left(3w-7\right)}{\left(3w-7\right)\left(3w+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{w\left(3w+7\right)+4\left(3w-7\right)}{\left(3w-7\right)\left(3w+7\right)}$$

Do the multiplications in w\left(3w+7\right)+4\left(3w-7\right).

$$\frac{3w^{2}+7w+12w-28}{\left(3w-7\right)\left(3w+7\right)}$$

Combine like terms in 3w^{2}+7w+12w-28.

$$\frac{3w^{2}+19w-28}{\left(3w-7\right)\left(3w+7\right)}$$

Expand \left(3w-7\right)\left(3w+7\right).

$$\frac{3w^{2}+19w-28}{9w^{2}-49}$$

Differentiate w.r.t. w

$$\frac{-171w^{2}+210w-931}{\left(9w^{2}-49\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3w-7 and 3w+7 is \left(3w-7\right)\left(3w+7\right). Multiply \frac{w}{3w-7} times \frac{3w+7}{3w+7}. Multiply \frac{4}{3w+7} times \frac{3w-7}{3w-7}.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{w\left(3w+7\right)}{\left(3w-7\right)\left(3w+7\right)}+\frac{4\left(3w-7\right)}{\left(3w-7\right)\left(3w+7\right)})$$

Since \frac{w\left(3w+7\right)}{\left(3w-7\right)\left(3w+7\right)} and \frac{4\left(3w-7\right)}{\left(3w-7\right)\left(3w+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{w\left(3w+7\right)+4\left(3w-7\right)}{\left(3w-7\right)\left(3w+7\right)})$$

Do the multiplications in w\left(3w+7\right)+4\left(3w-7\right).

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{3w^{2}+7w+12w-28}{\left(3w-7\right)\left(3w+7\right)})$$

Combine like terms in 3w^{2}+7w+12w-28.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{3w^{2}+19w-28}{\left(3w-7\right)\left(3w+7\right)})$$

Consider \left(3w-7\right)\left(3w+7\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{3w^{2}+19w-28}{\left(3w\right)^{2}-7^{2}})$$

Expand \left(3w\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{3w^{2}+19w-28}{3^{2}w^{2}-7^{2}})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{3w^{2}+19w-28}{9w^{2}-7^{2}})$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{3w^{2}+19w-28}{9w^{2}-49})$$

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.

$$\frac{\left(9w^{2}-49\right)\frac{\mathrm{d}}{\mathrm{d}w}(3w^{2}+19w^{1}-28)-\left(3w^{2}+19w^{1}-28\right)\frac{\mathrm{d}}{\mathrm{d}w}(9w^{2}-49)}{\left(9w^{2}-49\right)^{2}}$$

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}.

$$\frac{\left(9w^{2}-49\right)\left(2\times 3w^{2-1}+19w^{1-1}\right)-\left(3w^{2}+19w^{1}-28\right)\times 2\times 9w^{2-1}}{\left(9w^{2}-49\right)^{2}}$$

Simplify.

$$\frac{\left(9w^{2}-49\right)\left(6w^{1}+19w^{0}\right)-\left(3w^{2}+19w^{1}-28\right)\times 18w^{1}}{\left(9w^{2}-49\right)^{2}}$$

Multiply 9w^{2}-49 times 6w^{1}+19w^{0}.

$$\frac{9w^{2}\times 6w^{1}+9w^{2}\times 19w^{0}-49\times 6w^{1}-49\times 19w^{0}-\left(3w^{2}+19w^{1}-28\right)\times 18w^{1}}{\left(9w^{2}-49\right)^{2}}$$

Multiply 3w^{2}+19w^{1}-28 times 18w^{1}.

$$\frac{9w^{2}\times 6w^{1}+9w^{2}\times 19w^{0}-49\times 6w^{1}-49\times 19w^{0}-\left(3w^{2}\times 18w^{1}+19w^{1}\times 18w^{1}-28\times 18w^{1}\right)}{\left(9w^{2}-49\right)^{2}}$$

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

$$\frac{9\times 6w^{2+1}+9\times 19w^{2}-49\times 6w^{1}-49\times 19w^{0}-\left(3\times 18w^{2+1}+19\times 18w^{1+1}-28\times 18w^{1}\right)}{\left(9w^{2}-49\right)^{2}}$$

Simplify.

$$\frac{54w^{3}+171w^{2}-294w^{1}-931w^{0}-\left(54w^{3}+342w^{2}-504w^{1}\right)}{\left(9w^{2}-49\right)^{2}}$$

Combine like terms.

$$\frac{-171w^{2}+210w^{1}-931w^{0}}{\left(9w^{2}-49\right)^{2}}$$

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

$$\frac{-171w^{2}+210w-931w^{0}}{\left(9w^{2}-49\right)^{2}}$$

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

$$\frac{-171w^{2}+210w-931}{\left(9w^{2}-49\right)^{2}}$$