Solve 1-1/1-1{1-frac{1/a}}=?

$$1 - \frac { 1 } { 1 - \frac { 1 } { 1 - \frac { 1 } { a } } } = ?$$

Evaluate

$$a$$

Short Solution Steps

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

$$1-\frac{1}{1-\frac{1}{\frac{a}{a}-\frac{1}{a}}}$$

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

$$1-\frac{1}{1-\frac{1}{\frac{a-1}{a}}}$$

Divide 1 by \frac{a-1}{a} by multiplying 1 by the reciprocal of \frac{a-1}{a}.

$$1-\frac{1}{1-\frac{a}{a-1}}$$

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

$$1-\frac{1}{\frac{a-1}{a-1}-\frac{a}{a-1}}$$

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

$$1-\frac{1}{\frac{a-1-a}{a-1}}$$

Combine like terms in a-1-a.

$$1-\frac{1}{\frac{-1}{a-1}}$$

Divide 1 by \frac{-1}{a-1} by multiplying 1 by the reciprocal of \frac{-1}{a-1}.

$$1-\frac{a-1}{-1}$$

Anything divided by -1 gives its opposite. To find the opposite of a-1, find the opposite of each term.

$$1-\left(-a-\left(-1\right)\right)$$

To find the opposite of -a-\left(-1\right), find the opposite of each term.

$$1-\left(-a\right)-\left(-\left(-1\right)\right)$$

The opposite of -a is a.

$$1+a-\left(-\left(-1\right)\right)$$

The opposite of -1 is 1.

$$1+a-1$$

Subtract 1 from 1 to get 0.

$$a$$

Differentiate w.r.t. a

$$1$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{1}{1-\frac{1}{\frac{a}{a}-\frac{1}{a}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{1}{1-\frac{1}{\frac{a-1}{a}}})$$

Divide 1 by \frac{a-1}{a} by multiplying 1 by the reciprocal of \frac{a-1}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{1}{1-\frac{a}{a-1}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{1}{\frac{a-1}{a-1}-\frac{a}{a-1}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{1}{\frac{a-1-a}{a-1}})$$

Combine like terms in a-1-a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{1}{\frac{-1}{a-1}})$$

Divide 1 by \frac{-1}{a-1} by multiplying 1 by the reciprocal of \frac{-1}{a-1}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\frac{a-1}{-1})$$

Anything divided by -1 gives its opposite. To find the opposite of a-1, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\left(-a-\left(-1\right)\right))$$

To find the opposite of -a-\left(-1\right), find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1-\left(-a\right)-\left(-\left(-1\right)\right))$$

The opposite of -a is a.

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

The opposite of -1 is 1.

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

Subtract 1 from 1 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a)$$

The derivative of ax^{n} is nax^{n-1}.

$$a^{1-1}$$

Subtract 1 from 1.

$$a^{0}$$

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

$$1$$