Solve 3/x-3-7/5x-1

$$\frac{ 3 }{ x-3 } - \frac{ 7 }{ 5x-1 }$$

Evaluate

$$\frac{2\left(4x+9\right)}{\left(x-3\right)\left(5x-1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 5x-1 is \left(x-3\right)\left(5x-1\right). Multiply \frac{3}{x-3} times \frac{5x-1}{5x-1}. Multiply \frac{7}{5x-1} times \frac{x-3}{x-3}.

$$\frac{3\left(5x-1\right)}{\left(x-3\right)\left(5x-1\right)}-\frac{7\left(x-3\right)}{\left(x-3\right)\left(5x-1\right)}$$

Since \frac{3\left(5x-1\right)}{\left(x-3\right)\left(5x-1\right)} and \frac{7\left(x-3\right)}{\left(x-3\right)\left(5x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(5x-1\right)-7\left(x-3\right)}{\left(x-3\right)\left(5x-1\right)}$$

Do the multiplications in 3\left(5x-1\right)-7\left(x-3\right).

$$\frac{15x-3-7x+21}{\left(x-3\right)\left(5x-1\right)}$$

Combine like terms in 15x-3-7x+21.

$$\frac{8x+18}{\left(x-3\right)\left(5x-1\right)}$$

Expand \left(x-3\right)\left(5x-1\right).

$$\frac{8x+18}{5x^{2}-16x+3}$$

Differentiate w.r.t. x

$$\frac{4\left(78-45x-10x^{2}\right)}{25x^{4}-160x^{3}+286x^{2}-96x+9}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 5x-1 is \left(x-3\right)\left(5x-1\right). Multiply \frac{3}{x-3} times \frac{5x-1}{5x-1}. Multiply \frac{7}{5x-1} times \frac{x-3}{x-3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\left(5x-1\right)}{\left(x-3\right)\left(5x-1\right)}-\frac{7\left(x-3\right)}{\left(x-3\right)\left(5x-1\right)})$$

Since \frac{3\left(5x-1\right)}{\left(x-3\right)\left(5x-1\right)} and \frac{7\left(x-3\right)}{\left(x-3\right)\left(5x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\left(5x-1\right)-7\left(x-3\right)}{\left(x-3\right)\left(5x-1\right)})$$

Do the multiplications in 3\left(5x-1\right)-7\left(x-3\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{15x-3-7x+21}{\left(x-3\right)\left(5x-1\right)})$$

Combine like terms in 15x-3-7x+21.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8x+18}{\left(x-3\right)\left(5x-1\right)})$$

Apply the distributive property by multiplying each term of x-3 by each term of 5x-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8x+18}{5x^{2}-x-15x+3})$$

Combine -x and -15x to get -16x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8x+18}{5x^{2}-16x+3})$$

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.

$$\frac{\left(5x^{2}-16x^{1}+3\right)\frac{\mathrm{d}}{\mathrm{d}x}(8x^{1}+18)-\left(8x^{1}+18\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{2}-16x^{1}+3)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

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}.

$$\frac{\left(5x^{2}-16x^{1}+3\right)\times 8x^{1-1}-\left(8x^{1}+18\right)\left(2\times 5x^{2-1}-16x^{1-1}\right)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

Simplify.

$$\frac{\left(5x^{2}-16x^{1}+3\right)\times 8x^{0}-\left(8x^{1}+18\right)\left(10x^{1}-16x^{0}\right)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

Multiply 5x^{2}-16x^{1}+3 times 8x^{0}.

$$\frac{5x^{2}\times 8x^{0}-16x^{1}\times 8x^{0}+3\times 8x^{0}-\left(8x^{1}+18\right)\left(10x^{1}-16x^{0}\right)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

Multiply 8x^{1}+18 times 10x^{1}-16x^{0}.

$$\frac{5x^{2}\times 8x^{0}-16x^{1}\times 8x^{0}+3\times 8x^{0}-\left(8x^{1}\times 10x^{1}+8x^{1}\left(-16\right)x^{0}+18\times 10x^{1}+18\left(-16\right)x^{0}\right)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

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

$$\frac{5\times 8x^{2}-16\times 8x^{1}+3\times 8x^{0}-\left(8\times 10x^{1+1}+8\left(-16\right)x^{1}+18\times 10x^{1}+18\left(-16\right)x^{0}\right)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

Simplify.

$$\frac{40x^{2}-128x^{1}+24x^{0}-\left(80x^{2}-128x^{1}+180x^{1}-288x^{0}\right)}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

Combine like terms.

$$\frac{-40x^{2}-180x^{1}+312x^{0}}{\left(5x^{2}-16x^{1}+3\right)^{2}}$$

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

$$\frac{-40x^{2}-180x+312x^{0}}{\left(5x^{2}-16x+3\right)^{2}}$$

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

$$\frac{-40x^{2}-180x+312\times 1}{\left(5x^{2}-16x+3\right)^{2}}$$

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

$$\frac{-40x^{2}-180x+312}{\left(5x^{2}-16x+3\right)^{2}}$$