Solve 3x^2/4x^3

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 3 times -1.

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

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

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

Add the exponents 2 and -3.

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

Raise 3 to the power 1.

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

Multiply 3 times \frac{1}{4}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 3 from 2.

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

Divide 3 by 4.

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{4}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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