Solve 1x/2x-1-1/2x+1

$$\frac { 1 x } { 2 x - 1 } - \frac { 1 } { 2 x + 1 }$$

Evaluate

$$\frac{2x^{2}-x+1}{4x^{2}-1}$$

Short Solution Steps

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

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

Since \frac{1x\left(2x+1\right)}{\left(2x-1\right)\left(2x+1\right)} and \frac{2x-1}{\left(2x-1\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{2x^{2}+x-2x+1}{\left(2x-1\right)\left(2x+1\right)}$$

Combine like terms in 2x^{2}+x-2x+1.

$$\frac{2x^{2}-x+1}{\left(2x-1\right)\left(2x+1\right)}$$

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

$$\frac{2x^{2}-x+1}{4x^{2}-1}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

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

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

Since \frac{1x\left(2x+1\right)}{\left(2x-1\right)\left(2x+1\right)} and \frac{2x-1}{\left(2x-1\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}+x-2x+1}{\left(2x-1\right)\left(2x+1\right)})$$

Combine like terms in 2x^{2}+x-2x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}-x+1}{\left(2x-1\right)\left(2x+1\right)})$$

Consider \left(2x-1\right)\left(2x+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}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}-x+1}{\left(2x\right)^{2}-1^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}-x+1}{2^{2}x^{2}-1^{2}})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}-x+1}{4x^{2}-1^{2}})$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}-x+1}{4x^{2}-1})$$

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.

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

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}.

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

Simplify.

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

Multiply 4x^{2}-1 times 4x^{1}-x^{0}.

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

Multiply 2x^{2}-x^{1}+1 times 8x^{1}.

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

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

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

Simplify.

$$\frac{16x^{3}-4x^{2}-4x^{1}+x^{0}-\left(16x^{3}-8x^{2}+8x^{1}\right)}{\left(4x^{2}-1\right)^{2}}$$

Combine like terms.

$$\frac{4x^{2}-12x^{1}+x^{0}}{\left(4x^{2}-1\right)^{2}}$$

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

$$\frac{4x^{2}-12x+x^{0}}{\left(4x^{2}-1\right)^{2}}$$

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

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