Solve d/dxleft(10-5/x^3right)

$$\frac{d}{d x } \left(10- \frac{ 5 }{ { x }^{ 3 } } \right)$$

Evaluate

$$\frac{15}{x^{4}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{x^{3}}{x^{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{3}}{x^{3}}-\frac{5}{x^{3}})$$

Since \frac{10x^{3}}{x^{3}} and \frac{5}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{3}-5}{x^{3}})$$

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.

$$\frac{x^{3}\frac{\mathrm{d}}{\mathrm{d}x}(10x^{3}-5)-\left(10x^{3}-5\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{3})}{\left(x^{3}\right)^{2}}$$

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}.

$$\frac{x^{3}\times 3\times 10x^{3-1}-\left(10x^{3}-5\right)\times 3x^{3-1}}{\left(x^{3}\right)^{2}}$$

Do the arithmetic.

$$\frac{x^{3}\times 30x^{2}-\left(10x^{3}-5\right)\times 3x^{2}}{\left(x^{3}\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{3}\times 30x^{2}-\left(10x^{3}\times 3x^{2}-5\times 3x^{2}\right)}{\left(x^{3}\right)^{2}}$$

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

$$\frac{30x^{3+2}-\left(10\times 3x^{3+2}-5\times 3x^{2}\right)}{\left(x^{3}\right)^{2}}$$

Do the arithmetic.

$$\frac{30x^{5}-\left(30x^{5}-15x^{2}\right)}{\left(x^{3}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{30x^{5}-30x^{5}-\left(-15x^{2}\right)}{\left(x^{3}\right)^{2}}$$

Combine like terms.

$$\frac{\left(30-30\right)x^{5}+\left(-\left(-15\right)\right)x^{2}}{\left(x^{3}\right)^{2}}$$

Subtract 30 from 30.

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

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

$$-\frac{-15x^{2}}{x^{3\times 2}}$$

Multiply 3 times 2.

$$\frac{\left(-\left(-15\right)\right)x^{2}}{x^{6}}$$

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

$$\left(-\frac{-15}{1}\right)x^{2-6}$$

Do the arithmetic.

$$15x^{-4}$$

Differentiate w.r.t. x

$$-\frac{60}{x^{5}}$$