Solve 2x^-1/3x^-3

$$\frac { 2 x ^ { - 1 } } { 3 x ^ { - 3 } }$$

Evaluate

$$\frac{2x^{2}}{3}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply -3 times -1.

$$2^{1}\times \frac{1}{3}\times \frac{1}{x}x^{3}$$

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

$$2^{1}\times \frac{1}{3}x^{-1+3}$$

Add the exponents -1 and 3.

$$2^{1}\times \frac{1}{3}x^{2}$$

Raise 2 to the power 1.

$$2\times \frac{1}{3}x^{2}$$

Multiply 2 times \frac{1}{3}.

$$\frac{2}{3}x^{2}$$

Differentiate w.r.t. x

$$\frac{4x}{3}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3}x^{2})$$

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}.

$$2\times \frac{2}{3}x^{2-1}$$

Do the arithmetic.

$$\frac{4}{3}x^{1}$$

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

$$\frac{4}{3}x$$