Solve 15x/x-3

$$\frac { 15 x } { x - 3 }$$

Differentiate w.r.t. x

$$-\frac{45}{\left(x-3\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

$$\frac{15x^{1}-45x^{0}-15x^{1}}{\left(x^{1}-3\right)^{2}}$$

Combine like terms.

$$\frac{\left(15-15\right)x^{1}-45x^{0}}{\left(x^{1}-3\right)^{2}}$$

Subtract 15 from 15.

$$\frac{-45x^{0}}{\left(x^{1}-3\right)^{2}}$$

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

$$\frac{-45x^{0}}{\left(x-3\right)^{2}}$$

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

$$\frac{-45}{\left(x-3\right)^{2}}$$

Evaluate

$$\frac{15x}{x-3}$$