Solve d/dx5x^2-6x+1/x^4-x^3

$$\frac { d } { d x } \frac { 5 x ^ { 2 } - 6 x + 1 } { x ^ { 4 } - x ^ { 3 } }$$

Evaluate

$$\frac{3-10x}{x^{4}}$$

Steps Using Derivative Rule for Quotient

Factor the expressions that are not already factored in \frac{5x^{2}-6x+1}{x^{4}-x^{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-1\right)\left(5x-1\right)}{\left(x-1\right)x^{3}})$$

Cancel out x-1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x-1}{x^{3}})$$

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.

$$\frac{x^{3}\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}-1)-\left(5x^{1}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{3})}{\left(x^{3}\right)^{2}}$$

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}.

$$\frac{x^{3}\times 5x^{1-1}-\left(5x^{1}-1\right)\times 3x^{3-1}}{\left(x^{3}\right)^{2}}$$

Do the arithmetic.

$$\frac{x^{3}\times 5x^{0}-\left(5x^{1}-1\right)\times 3x^{2}}{\left(x^{3}\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{3}\times 5x^{0}-\left(5x^{1}\times 3x^{2}-3x^{2}\right)}{\left(x^{3}\right)^{2}}$$

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

$$\frac{5x^{3}-\left(5\times 3x^{1+2}-3x^{2}\right)}{\left(x^{3}\right)^{2}}$$

Do the arithmetic.

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

Remove unnecessary parentheses.

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

Combine like terms.

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

Subtract 15 from 5.

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

Factor out x^{2}.

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

To raise a power to another power, multiply the exponents.

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

Multiply 3 times 2.

$$\frac{x^{2}\left(-10x^{1}-\left(-3x^{0}\right)\right)}{x^{6}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{-10x^{1}-\left(-3x^{0}\right)}{x^{6-2}}$$

Subtract 2 from 6.

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

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

$$\frac{-10x-\left(-3x^{0}\right)}{x^{4}}$$

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

$$\frac{-10x-\left(-3\right)}{x^{4}}$$

Differentiate w.r.t. x

$$\frac{6\left(5x-2\right)}{x^{5}}$$