Solve d/dxleft(3x^2-2/x-5right)562

$$\frac{d}{d x } \left( \frac{ 3 { x }^{ 2 } -2 }{ x-5 } \right) 562$$

Evaluate

$$\frac{562\left(3x^{2}-30x+2\right)}{\left(x-5\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express \frac{3x^{2}-2}{x-5}\times 562 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(3x^{2}-2\right)\times 562}{x-5})$$

Use the distributive property to multiply 3x^{2}-2 by 562.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1686x^{2}-1124}{x-5})$$

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.

$$\frac{\left(x^{1}-5\right)\frac{\mathrm{d}}{\mathrm{d}x}(1686x^{2}-1124)-\left(1686x^{2}-1124\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-5)}{\left(x^{1}-5\right)^{2}}$$

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}.

$$\frac{\left(x^{1}-5\right)\times 2\times 1686x^{2-1}-\left(1686x^{2}-1124\right)x^{1-1}}{\left(x^{1}-5\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(x^{1}-5\right)\times 3372x^{1}-\left(1686x^{2}-1124\right)x^{0}}{\left(x^{1}-5\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 3372x^{1}-5\times 3372x^{1}-\left(1686x^{2}x^{0}-1124x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

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

$$\frac{3372x^{1+1}-5\times 3372x^{1}-\left(1686x^{2}-1124x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Do the arithmetic.

$$\frac{3372x^{2}-16860x^{1}-\left(1686x^{2}-1124x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{3372x^{2}-16860x^{1}-1686x^{2}-\left(-1124x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Combine like terms.

$$\frac{\left(3372-1686\right)x^{2}-16860x^{1}-\left(-1124x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

Subtract 1686 from 3372.

$$\frac{1686x^{2}-16860x^{1}-\left(-1124x^{0}\right)}{\left(x^{1}-5\right)^{2}}$$

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

$$\frac{1686x^{2}-16860x-\left(-1124x^{0}\right)}{\left(x-5\right)^{2}}$$

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

$$\frac{1686x^{2}-16860x-\left(-1124\right)}{\left(x-5\right)^{2}}$$

Differentiate w.r.t. x

$$\frac{82052}{\left(x-5\right)^{3}}$$