Solve (9/(4x-1+x))div20x

$$( \frac{ 9 }{ (4x-1+x) } ) \div 20x$$

Evaluate

$$\frac{9x}{20\left(5x-1\right)}$$

Short Solution Steps

Express \frac{\frac{9}{4x-1+x}}{20} as a single fraction.

$$\frac{9}{\left(4x-1+x\right)\times 20}x$$

Combine 4x and x to get 5x.

$$\frac{9}{\left(5x-1\right)\times 20}x$$

Express \frac{9}{\left(5x-1\right)\times 20}x as a single fraction.

$$\frac{9x}{\left(5x-1\right)\times 20}$$

Use the distributive property to multiply 5x-1 by 20.

$$\frac{9x}{100x-20}$$

Differentiate w.r.t. x

$$-\frac{9}{20\left(5x-1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express \frac{\frac{9}{4x-1+x}}{20} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9}{\left(4x-1+x\right)\times 20}x)$$

Combine 4x and x to get 5x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9}{\left(5x-1\right)\times 20}x)$$

Express \frac{9}{\left(5x-1\right)\times 20}x as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x}{\left(5x-1\right)\times 20})$$

Use the distributive property to multiply 5x-1 by 20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x}{100x-20})$$

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.

$$\frac{\left(100x^{1}-20\right)\frac{\mathrm{d}}{\mathrm{d}x}(9x^{1})-9x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(100x^{1}-20)}{\left(100x^{1}-20\right)^{2}}$$

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}.

$$\frac{\left(100x^{1}-20\right)\times 9x^{1-1}-9x^{1}\times 100x^{1-1}}{\left(100x^{1}-20\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(100x^{1}-20\right)\times 9x^{0}-9x^{1}\times 100x^{0}}{\left(100x^{1}-20\right)^{2}}$$

Expand using distributive property.

$$\frac{100x^{1}\times 9x^{0}-20\times 9x^{0}-9x^{1}\times 100x^{0}}{\left(100x^{1}-20\right)^{2}}$$

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

$$\frac{100\times 9x^{1}-20\times 9x^{0}-9\times 100x^{1}}{\left(100x^{1}-20\right)^{2}}$$

Do the arithmetic.

$$\frac{900x^{1}-180x^{0}-900x^{1}}{\left(100x^{1}-20\right)^{2}}$$

Combine like terms.

$$\frac{\left(900-900\right)x^{1}-180x^{0}}{\left(100x^{1}-20\right)^{2}}$$

Subtract 900 from 900.

$$\frac{-180x^{0}}{\left(100x^{1}-20\right)^{2}}$$

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

$$\frac{-180x^{0}}{\left(100x-20\right)^{2}}$$

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

$$\frac{-180}{\left(100x-20\right)^{2}}$$