Solve 5(2x/x+3)

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

Evaluate

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

Solution Steps

Express 5\times \frac{2x}{x+3} as a single fraction.

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

Multiply 5 and 2 to get 10.

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

Differentiate w.r.t. x

$$\frac{30}{\left(x+3\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express 5\times \frac{2x}{x+3} as a single fraction.

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

Multiply 5 and 2 to get 10.

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

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

$$\frac{10x^{1}+30x^{0}-10x^{1}}{\left(x^{1}+3\right)^{2}}$$

Combine like terms.

$$\frac{\left(10-10\right)x^{1}+30x^{0}}{\left(x^{1}+3\right)^{2}}$$

Subtract 10 from 10.

$$\frac{30x^{0}}{\left(x^{1}+3\right)^{2}}$$

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

$$\frac{30x^{0}}{\left(x+3\right)^{2}}$$

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

$$\frac{30\times 1}{\left(x+3\right)^{2}}$$

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

$$\frac{30}{\left(x+3\right)^{2}}$$