Solve 3x^2/9x^4

$$\frac { 3 x ^ { 2 } } { 9 x ^ { 4 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(3x^{2}\right)^{1}\times \frac{1}{9x^{4}}$$

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

$$3^{1}\left(x^{2}\right)^{1}\times \frac{1}{9}\times \frac{1}{x^{4}}$$

Use the Commutative Property of Multiplication.

$$3^{1}\times \frac{1}{9}\left(x^{2}\right)^{1}\times \frac{1}{x^{4}}$$

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

$$3^{1}\times \frac{1}{9}x^{2}x^{4\left(-1\right)}$$

Multiply 4 times -1.

$$3^{1}\times \frac{1}{9}x^{2}x^{-4}$$

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

$$3^{1}\times \frac{1}{9}x^{2-4}$$

Add the exponents 2 and -4.

$$3^{1}\times \frac{1}{9}x^{-2}$$

Raise 3 to the power 1.

$$3\times \frac{1}{9}x^{-2}$$

Multiply 3 times \frac{1}{9}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}x^{2}}{9^{1}x^{4}}$$

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

$$\frac{3^{1}x^{2-4}}{9^{1}}$$

Subtract 4 from 2.

$$\frac{3^{1}x^{-2}}{9^{1}}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{9}x^{2-4})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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