Solve 1/5x-3-3/2+3x

$$\frac { 1 } { 5 x - 3 } - \frac { 3 } { 2 + 3 x }$$

Evaluate

$$\frac{11-12x}{\left(5x-3\right)\left(3x+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5x-3 and 2+3x is \left(5x-3\right)\left(3x+2\right). Multiply \frac{1}{5x-3} times \frac{3x+2}{3x+2}. Multiply \frac{3}{2+3x} times \frac{5x-3}{5x-3}.

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

Since \frac{3x+2}{\left(5x-3\right)\left(3x+2\right)} and \frac{3\left(5x-3\right)}{\left(5x-3\right)\left(3x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x+2-3\left(5x-3\right).

$$\frac{3x+2-15x+9}{\left(5x-3\right)\left(3x+2\right)}$$

Combine like terms in 3x+2-15x+9.

$$\frac{-12x+11}{\left(5x-3\right)\left(3x+2\right)}$$

Expand \left(5x-3\right)\left(3x+2\right).

$$\frac{-12x+11}{15x^{2}+x-6}$$

Differentiate w.r.t. x

$$\frac{180x^{2}-330x+61}{225x^{4}+30x^{3}-179x^{2}-12x+36}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5x-3 and 2+3x is \left(5x-3\right)\left(3x+2\right). Multiply \frac{1}{5x-3} times \frac{3x+2}{3x+2}. Multiply \frac{3}{2+3x} times \frac{5x-3}{5x-3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+2}{\left(5x-3\right)\left(3x+2\right)}-\frac{3\left(5x-3\right)}{\left(5x-3\right)\left(3x+2\right)})$$

Since \frac{3x+2}{\left(5x-3\right)\left(3x+2\right)} and \frac{3\left(5x-3\right)}{\left(5x-3\right)\left(3x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+2-3\left(5x-3\right)}{\left(5x-3\right)\left(3x+2\right)})$$

Do the multiplications in 3x+2-3\left(5x-3\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+2-15x+9}{\left(5x-3\right)\left(3x+2\right)})$$

Combine like terms in 3x+2-15x+9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-12x+11}{\left(5x-3\right)\left(3x+2\right)})$$

Apply the distributive property by multiplying each term of 5x-3 by each term of 3x+2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-12x+11}{15x^{2}+10x-9x-6})$$

Combine 10x and -9x to get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-12x+11}{15x^{2}+x-6})$$

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.

$$\frac{\left(15x^{2}+x^{1}-6\right)\frac{\mathrm{d}}{\mathrm{d}x}(-12x^{1}+11)-\left(-12x^{1}+11\right)\frac{\mathrm{d}}{\mathrm{d}x}(15x^{2}+x^{1}-6)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

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}.

$$\frac{\left(15x^{2}+x^{1}-6\right)\left(-12\right)x^{1-1}-\left(-12x^{1}+11\right)\left(2\times 15x^{2-1}+x^{1-1}\right)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

Simplify.

$$\frac{\left(15x^{2}+x^{1}-6\right)\left(-12\right)x^{0}-\left(-12x^{1}+11\right)\left(30x^{1}+x^{0}\right)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

Multiply 15x^{2}+x^{1}-6 times -12x^{0}.

$$\frac{15x^{2}\left(-12\right)x^{0}+x^{1}\left(-12\right)x^{0}-6\left(-12\right)x^{0}-\left(-12x^{1}+11\right)\left(30x^{1}+x^{0}\right)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

Multiply -12x^{1}+11 times 30x^{1}+x^{0}.

$$\frac{15x^{2}\left(-12\right)x^{0}+x^{1}\left(-12\right)x^{0}-6\left(-12\right)x^{0}-\left(-12x^{1}\times 30x^{1}-12x^{1}x^{0}+11\times 30x^{1}+11x^{0}\right)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

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

$$\frac{15\left(-12\right)x^{2}-12x^{1}-6\left(-12\right)x^{0}-\left(-12\times 30x^{1+1}-12x^{1}+11\times 30x^{1}+11x^{0}\right)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

Simplify.

$$\frac{-180x^{2}-12x^{1}+72x^{0}-\left(-360x^{2}-12x^{1}+330x^{1}+11x^{0}\right)}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

Combine like terms.

$$\frac{180x^{2}-330x^{1}+61x^{0}}{\left(15x^{2}+x^{1}-6\right)^{2}}$$

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

$$\frac{180x^{2}-330x+61x^{0}}{\left(15x^{2}+x-6\right)^{2}}$$

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

$$\frac{180x^{2}-330x+61\times 1}{\left(15x^{2}+x-6\right)^{2}}$$

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

$$\frac{180x^{2}-330x+61}{\left(15x^{2}+x-6\right)^{2}}$$