Solve 1-1/(2n+1)

$$1 - \frac { 1 } { ( 2 n + 1 ) }$$

Evaluate

$$\frac{2n}{2n+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2n+1}{2n+1}.

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

Since \frac{2n+1}{2n+1} and \frac{1}{2n+1} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in 2n+1-1.

$$\frac{2n}{2n+1}$$

Differentiate w.r.t. n

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

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2n+1}{2n+1}.

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

Since \frac{2n+1}{2n+1} and \frac{1}{2n+1} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in 2n+1-1.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{2n}{2n+1})$$

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

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

Combine like terms.

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

Subtract 4 from 4.

$$\frac{2n^{0}}{\left(2n^{1}+1\right)^{2}}$$

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

$$\frac{2n^{0}}{\left(2n+1\right)^{2}}$$

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

$$\frac{2\times 1}{\left(2n+1\right)^{2}}$$

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

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