Solve 3x/2x+5div2x/x+5

$$\frac { 3 x } { 2 x + 5 } \div \frac { 2 x } { x + 5 }$$

Evaluate

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

Short Solution Steps

Divide \frac{3x}{2x+5} by \frac{2x}{x+5} by multiplying \frac{3x}{2x+5} by the reciprocal of \frac{2x}{x+5}.

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

Cancel out x in both numerator and denominator.

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

Use the distributive property to multiply 3 by x+5.

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

Use the distributive property to multiply 2 by 2x+5.

$$\frac{3x+15}{4x+10}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Divide \frac{3x}{2x+5} by \frac{2x}{x+5} by multiplying \frac{3x}{2x+5} by the reciprocal of \frac{2x}{x+5}.

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

Cancel out x in both numerator and denominator.

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

Use the distributive property to multiply 3 by x+5.

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

Use the distributive property to multiply 2 by 2x+5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+15}{4x+10})$$

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.

$$\frac{\left(4x^{1}+10\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{1}+15)-\left(3x^{1}+15\right)\frac{\mathrm{d}}{\mathrm{d}x}(4x^{1}+10)}{\left(4x^{1}+10\right)^{2}}$$

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}.

$$\frac{\left(4x^{1}+10\right)\times 3x^{1-1}-\left(3x^{1}+15\right)\times 4x^{1-1}}{\left(4x^{1}+10\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(4x^{1}+10\right)\times 3x^{0}-\left(3x^{1}+15\right)\times 4x^{0}}{\left(4x^{1}+10\right)^{2}}$$

Expand using distributive property.

$$\frac{4x^{1}\times 3x^{0}+10\times 3x^{0}-\left(3x^{1}\times 4x^{0}+15\times 4x^{0}\right)}{\left(4x^{1}+10\right)^{2}}$$

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

$$\frac{4\times 3x^{1}+10\times 3x^{0}-\left(3\times 4x^{1}+15\times 4x^{0}\right)}{\left(4x^{1}+10\right)^{2}}$$

Do the arithmetic.

$$\frac{12x^{1}+30x^{0}-\left(12x^{1}+60x^{0}\right)}{\left(4x^{1}+10\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{12x^{1}+30x^{0}-12x^{1}-60x^{0}}{\left(4x^{1}+10\right)^{2}}$$

Combine like terms.

$$\frac{\left(12-12\right)x^{1}+\left(30-60\right)x^{0}}{\left(4x^{1}+10\right)^{2}}$$

Subtract 12 from 12 and 60 from 30.

$$\frac{-30x^{0}}{\left(4x^{1}+10\right)^{2}}$$

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

$$\frac{-30x^{0}}{\left(4x+10\right)^{2}}$$

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

$$\frac{-30}{\left(4x+10\right)^{2}}$$