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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in $x-x^{2}-x$.

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

Divide $1$ by $\frac{-x^{2}}{1-x}$ by multiplying $1$ by the reciprocal of $\frac{-x^{2}}{1-x}$.

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

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

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

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

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

Do the multiplications in $x\left(-1\right)x^{2}-\left(1-x\right)$.

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

Divide $x$ by $\frac{-x^{3}-1+x}{-x^{2}}$ by multiplying $x$ by the reciprocal of $\frac{-x^{3}-1+x}{-x^{2}}$.

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

To multiply powers of the same base, add their exponents. Add $1$ and $2$ to get $3$.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

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

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

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

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

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

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

Combine like terms in $x-x^{2}-x$.

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

Divide $1$ by $\frac{-x^{2}}{1-x}$ by multiplying $1$ by the reciprocal of $\frac{-x^{2}}{1-x}$.

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

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

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

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

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

Do the multiplications in $x\left(-1\right)x^{2}-\left(1-x\right)$.

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

Divide $x$ by $\frac{-x^{3}-1+x}{-x^{2}}$ by multiplying $x$ by the reciprocal of $\frac{-x^{3}-1+x}{-x^{2}}$.

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

To multiply powers of the same base, add their exponents. Add $1$ and $2$ to get $3$.

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

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.

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

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 $nax^{n-1}$.

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

Simplify.

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

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

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

Multiply $-x^{3}$ times $-3x^{2}+x^{0}$.

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

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

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

Simplify.

$$\frac{3x^{5}-3x^{3}+3x^{2}-\left(3x^{5}-x^{3}\right)}{\left(-x^{3}+x^{1}-1\right)^{2}}$$

Combine like terms.

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

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

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