Solve d/dxleft(3x^2-2/x-5right)284

$$\frac{d}{d x } \left( \frac{ 3 { x }^{ 2 } -2 }{ x-5 } \right) 284$$

Evaluate

$$\frac{284\left(3x^{2}-30x+2\right)}{\left(x-5\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express \frac{3x^{2}-2}{x-5}\times 284 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(3x^{2}-2\right)\times 284}{x-5})$$

Use the distributive property to multiply 3x^{2}-2 by 284.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{852x^{2}-568}{x-5})$$

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.

$$\frac{\left(x^{1}-5\right)\frac{\mathrm{d}}{\mathrm{d}x}(852x^{2}-568)-\left(852x^{2}-568\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-5)}{\left(x^{1}-5\right)^{2}}$$

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}.

$$\frac{\left(x^{1}-5\right)\times 2\times 852x^{2-1}-\left(852x^{2}-568\right)x^{1-1}}{\left(x^{1}-5\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(x^{1}-5\right)\times 1704x^{1}-\left(852x^{2}-568\right)x^{0}}{\left(x^{1}-5\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 1704x^{1}-5\times 1704x^{1}-\left(852x^{2}x^{0}-568x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

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

$$\frac{1704x^{1+1}-5\times 1704x^{1}-\left(852x^{2}-568x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Do the arithmetic.

$$\frac{1704x^{2}-8520x^{1}-\left(852x^{2}-568x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{1704x^{2}-8520x^{1}-852x^{2}-\left(-568x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Combine like terms.

$$\frac{\left(1704-852\right)x^{2}-8520x^{1}-\left(-568x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Subtract 852 from 1704.

$$\frac{852x^{2}-8520x^{1}-\left(-568x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

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

$$\frac{852x^{2}-8520x-\left(-568x^{0}\right)}{\left(x-5\right)^{2}}$$

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

$$\frac{852x^{2}-8520x-\left(-568\right)}{\left(x-5\right)^{2}}$$

Differentiate w.r.t. x

$$\frac{41464}{\left(x-5\right)^{3}}$$