Solve 2+1/1-frac{1{x+1}}

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

Evaluate

$$3+\frac{1}{x}$$

Short Solution Steps

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

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

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

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

Combine like terms in x+1-1.

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

Divide 1 by \frac{x}{x+1} by multiplying 1 by the reciprocal of \frac{x}{x+1}.

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

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

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

Since \frac{2x}{x} and \frac{x+1}{x} have the same denominator, add them by adding their numerators.

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

Combine like terms in 2x+x+1.

$$\frac{3x+1}{x}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Product

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

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

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

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

Combine like terms in x+1-1.

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

Divide 1 by \frac{x}{x+1} by multiplying 1 by the reciprocal of \frac{x}{x+1}.

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

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

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

Since \frac{2x}{x} and \frac{x+1}{x} have the same denominator, add them by adding their numerators.

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

Combine like terms in 2x+x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+1}{x})$$

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

$$\left(3x^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x})+\frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x}(3x^{1}+1)$$

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}.

$$\left(3x^{1}+1\right)\left(-1\right)x^{-1-1}+\frac{1}{x}\times 3x^{1-1}$$

Simplify.

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

Multiply 3x^{1}+1 times -x^{-2}.

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

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

$$-3x^{1-2}-x^{-2}+3\times \frac{1}{x}$$

Simplify.

$$-3\times \frac{1}{x}-x^{-2}+3\times \frac{1}{x}$$

Steps Using Derivative Rule for Quotient

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

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

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

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

Combine like terms in x+1-1.

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

Divide 1 by \frac{x}{x+1} by multiplying 1 by the reciprocal of \frac{x}{x+1}.

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

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

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

Since \frac{2x}{x} and \frac{x+1}{x} have the same denominator, add them by adding their numerators.

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

Combine like terms in 2x+x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+1}{x})$$

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.

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

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}.

$$\frac{x^{1}\times 3x^{1-1}-\left(3x^{1}+1\right)x^{1-1}}{\left(x^{1}\right)^{2}}$$

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Remove unnecessary parentheses.

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

Combine like terms.

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

Subtract 3 from 3.

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

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$-\frac{x^{0}}{1^{2}x^{2}}$$

Raise 1 to the power 2.

$$-\frac{x^{0}}{x^{2}}$$

Multiply 1 times 2.

$$\frac{-x^{0}}{x^{2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Do the arithmetic.

$$-x^{-2}$$