Solve 5/x/frac{x^2{6}}

$$\frac { \frac { 5 } { x } } { \frac { x ^ { 2 } } { 6 } }$$

Evaluate

$$\frac{30}{x^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(5\times \frac{1}{x}\right)^{1}\times \frac{1}{\frac{1}{6}x^{2}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$5^{1}\times \left(\frac{1}{x}\right)^{1}\times \frac{1}{\frac{1}{6}}\times \frac{1}{x^{2}}$$

Use the Commutative Property of Multiplication.

$$5^{1}\times \frac{1}{\frac{1}{6}}\times \left(\frac{1}{x}\right)^{1}\times \frac{1}{x^{2}}$$

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

$$5^{1}\times \frac{1}{\frac{1}{6}}\times \frac{1}{x}x^{2\left(-1\right)}$$

Multiply 2 times -1.

$$5^{1}\times \frac{1}{\frac{1}{6}}\times \frac{1}{x}x^{-2}$$

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

$$5^{1}\times \frac{1}{\frac{1}{6}}x^{-1-2}$$

Add the exponents -1 and -2.

$$5^{1}\times \frac{1}{\frac{1}{6}}x^{-3}$$

Raise 5 to the power 1.

$$5\times \frac{1}{\frac{1}{6}}x^{-3}$$

Raise \frac{1}{6} to the power -1.

$$5\times 6x^{-3}$$

Multiply 5 times 6.

$$30x^{-3}$$

Differentiate w.r.t. x

$$-\frac{90}{x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{\frac{1}{6}}x^{-1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(30x^{-3})$$

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}.

$$-3\times 30x^{-3-1}$$

Do the arithmetic.

$$-90x^{-4}$$