Solve x^-4/3x^-3

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply -3 times -1.

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

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

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

Add the exponents -4 and 3.

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

Raise 3 to the power -1.

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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