Solve d/dxleft(3/x^3-4/3x^2-xright)

$$\frac{d}{d x } \left( \frac{ 3 }{ { x }^{ 3 } } - \frac{ 4 }{ 3 { x }^{ 2 } } -x \right)$$

Evaluate

$$-1+\frac{8}{3x^{3}}-\frac{9}{x^{4}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and 3x^{2} is 3x^{3}. Multiply \frac{3}{x^{3}} times \frac{3}{3}. Multiply \frac{4}{3x^{2}} times \frac{x}{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\times 3}{3x^{3}}-\frac{4x}{3x^{3}}-x)$$

Since \frac{3\times 3}{3x^{3}} and \frac{4x}{3x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\times 3-4x}{3x^{3}}-x)$$

Do the multiplications in 3\times 3-4x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9-4x}{3x^{3}}-x)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{3x^{3}}{3x^{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9-4x}{3x^{3}}-\frac{x\times 3x^{3}}{3x^{3}})$$

Since \frac{9-4x}{3x^{3}} and \frac{x\times 3x^{3}}{3x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9-4x-x\times 3x^{3}}{3x^{3}})$$

Do the multiplications in 9-4x-x\times 3x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9-4x-3x^{4}}{3x^{3}})$$

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.

$$\frac{3x^{3}\frac{\mathrm{d}}{\mathrm{d}x}(-4x^{1}-3x^{4}+9)-\left(-4x^{1}-3x^{4}+9\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3})}{\left(3x^{3}\right)^{2}}$$

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}.

$$\frac{3x^{3}\left(-4x^{1-1}+4\left(-3\right)x^{4-1}\right)-\left(-4x^{1}-3x^{4}+9\right)\times 3\times 3x^{3-1}}{\left(3x^{3}\right)^{2}}$$

Simplify.

$$\frac{3x^{3}\left(-4x^{0}-12x^{3}\right)-\left(-4x^{1}-3x^{4}+9\right)\times 9x^{2}}{\left(3x^{3}\right)^{2}}$$

Multiply 3x^{3} times -4x^{0}-12x^{3}.

$$\frac{3x^{3}\left(-4\right)x^{0}+3x^{3}\left(-12\right)x^{3}-\left(-4x^{1}-3x^{4}+9\right)\times 9x^{2}}{\left(3x^{3}\right)^{2}}$$

Multiply -4x^{1}-3x^{4}+9 times 9x^{2}.

$$\frac{3x^{3}\left(-4\right)x^{0}+3x^{3}\left(-12\right)x^{3}-\left(-4x^{1}\times 9x^{2}-3x^{4}\times 9x^{2}+9\times 9x^{2}\right)}{\left(3x^{3}\right)^{2}}$$

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

$$\frac{3\left(-4\right)x^{3}+3\left(-12\right)x^{3+3}-\left(-4\times 9x^{1+2}-3\times 9x^{4+2}+9\times 9x^{2}\right)}{\left(3x^{3}\right)^{2}}$$

Simplify.

$$\frac{-12x^{3}-36x^{6}-\left(-36x^{3}-27x^{6}+81x^{2}\right)}{\left(3x^{3}\right)^{2}}$$

Combine like terms.

$$\frac{24x^{3}-9x^{6}-81x^{2}}{\left(3x^{3}\right)^{2}}$$

Differentiate w.r.t. x

$$\frac{4\left(9-2x\right)}{x^{5}}$$