Solve -2x/4x^3

$$\frac{ -2x }{ 4 { x }^{ 3 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 3 times -1.

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

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

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

Add the exponents 1 and -3.

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

Raise -2 to the power 1.

$$-2\times \frac{1}{4}x^{-2}$$

Multiply -2 times \frac{1}{4}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 3 from 1.

$$\frac{\left(-2\right)^{1}x^{-2}}{4^{1}}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$x^{-3}$$