Solve 2/2f-1+5/2f+1

$$\frac { 2 } { 2 f - 1 } + \frac { 5 } { 2 f + 1 }$$

Evaluate

$$\frac{14f-3}{4f^{2}-1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2f-1 and 2f+1 is \left(2f-1\right)\left(2f+1\right). Multiply \frac{2}{2f-1} times \frac{2f+1}{2f+1}. Multiply \frac{5}{2f+1} times \frac{2f-1}{2f-1}.

$$\frac{2\left(2f+1\right)}{\left(2f-1\right)\left(2f+1\right)}+\frac{5\left(2f-1\right)}{\left(2f-1\right)\left(2f+1\right)}$$

Since \frac{2\left(2f+1\right)}{\left(2f-1\right)\left(2f+1\right)} and \frac{5\left(2f-1\right)}{\left(2f-1\right)\left(2f+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(2f+1\right)+5\left(2f-1\right)}{\left(2f-1\right)\left(2f+1\right)}$$

Do the multiplications in 2\left(2f+1\right)+5\left(2f-1\right).

$$\frac{4f+2+10f-5}{\left(2f-1\right)\left(2f+1\right)}$$

Combine like terms in 4f+2+10f-5.

$$\frac{14f-3}{\left(2f-1\right)\left(2f+1\right)}$$

Expand \left(2f-1\right)\left(2f+1\right).

$$\frac{14f-3}{4f^{2}-1}$$

Differentiate w.r.t. f

$$\frac{2\left(-28f^{2}+12f-7\right)}{\left(4f^{2}-1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2f-1 and 2f+1 is \left(2f-1\right)\left(2f+1\right). Multiply \frac{2}{2f-1} times \frac{2f+1}{2f+1}. Multiply \frac{5}{2f+1} times \frac{2f-1}{2f-1}.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{2\left(2f+1\right)}{\left(2f-1\right)\left(2f+1\right)}+\frac{5\left(2f-1\right)}{\left(2f-1\right)\left(2f+1\right)})$$

Since \frac{2\left(2f+1\right)}{\left(2f-1\right)\left(2f+1\right)} and \frac{5\left(2f-1\right)}{\left(2f-1\right)\left(2f+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{2\left(2f+1\right)+5\left(2f-1\right)}{\left(2f-1\right)\left(2f+1\right)})$$

Do the multiplications in 2\left(2f+1\right)+5\left(2f-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{4f+2+10f-5}{\left(2f-1\right)\left(2f+1\right)})$$

Combine like terms in 4f+2+10f-5.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{14f-3}{\left(2f-1\right)\left(2f+1\right)})$$

Consider \left(2f-1\right)\left(2f+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{14f-3}{\left(2f\right)^{2}-1^{2}})$$

Expand \left(2f\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{14f-3}{2^{2}f^{2}-1^{2}})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{14f-3}{4f^{2}-1^{2}})$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{14f-3}{4f^{2}-1})$$

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.

$$\frac{\left(4f^{2}-1\right)\frac{\mathrm{d}}{\mathrm{d}f}(14f^{1}-3)-\left(14f^{1}-3\right)\frac{\mathrm{d}}{\mathrm{d}f}(4f^{2}-1)}{\left(4f^{2}-1\right)^{2}}$$

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}.

$$\frac{\left(4f^{2}-1\right)\times 14f^{1-1}-\left(14f^{1}-3\right)\times 2\times 4f^{2-1}}{\left(4f^{2}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(4f^{2}-1\right)\times 14f^{0}-\left(14f^{1}-3\right)\times 8f^{1}}{\left(4f^{2}-1\right)^{2}}$$

Expand using distributive property.

$$\frac{4f^{2}\times 14f^{0}-14f^{0}-\left(14f^{1}\times 8f^{1}-3\times 8f^{1}\right)}{\left(4f^{2}-1\right)^{2}}$$

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

$$\frac{4\times 14f^{2}-14f^{0}-\left(14\times 8f^{1+1}-3\times 8f^{1}\right)}{\left(4f^{2}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{56f^{2}-14f^{0}-\left(112f^{2}-24f^{1}\right)}{\left(4f^{2}-1\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{56f^{2}-14f^{0}-112f^{2}-\left(-24f^{1}\right)}{\left(4f^{2}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(56-112\right)f^{2}-14f^{0}-\left(-24f^{1}\right)}{\left(4f^{2}-1\right)^{2}}$$

Subtract 112 from 56.

$$\frac{-56f^{2}-14f^{0}-\left(-24f^{1}\right)}{\left(4f^{2}-1\right)^{2}}$$

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

$$\frac{-56f^{2}-14f^{0}-\left(-24f\right)}{\left(4f^{2}-1\right)^{2}}$$

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

$$\frac{-56f^{2}-14-\left(-24f\right)}{\left(4f^{2}-1\right)^{2}}$$