Solve 12x^2/36x^3

$$\frac{ 12 { x }^{ 2 } }{ 36 { x }^{ 3 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(12x^{2}\right)^{1}\times \frac{1}{36x^{3}}$$

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

$$12^{1}\left(x^{2}\right)^{1}\times \frac{1}{36}\times \frac{1}{x^{3}}$$

Use the Commutative Property of Multiplication.

$$12^{1}\times \frac{1}{36}\left(x^{2}\right)^{1}\times \frac{1}{x^{3}}$$

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

$$12^{1}\times \frac{1}{36}x^{2}x^{3\left(-1\right)}$$

Multiply 3 times -1.

$$12^{1}\times \frac{1}{36}x^{2}x^{-3}$$

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

$$12^{1}\times \frac{1}{36}x^{2-3}$$

Add the exponents 2 and -3.

$$12^{1}\times \frac{1}{36}\times \frac{1}{x}$$

Raise 12 to the power 1.

$$12\times \frac{1}{36}\times \frac{1}{x}$$

Multiply 12 times \frac{1}{36}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{12^{1}x^{2}}{36^{1}x^{3}}$$

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

$$\frac{12^{1}x^{2-3}}{36^{1}}$$

Subtract 3 from 2.

$$\frac{12^{1}\times \frac{1}{x}}{36^{1}}$$

Reduce the fraction \frac{12}{36} to lowest terms by extracting and canceling out 12.

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

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.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12}{36}x^{2-3})$$

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.

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