Solve 9x/12x^3

$$\frac { 9 x } { 12 x ^ { 3 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(9x^{1}\right)^{1}\times \frac{1}{12x^{3}}$$

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

$$9^{1}\left(x^{1}\right)^{1}\times \frac{1}{12}\times \frac{1}{x^{3}}$$

Use the Commutative Property of Multiplication.

$$9^{1}\times \frac{1}{12}\left(x^{1}\right)^{1}\times \frac{1}{x^{3}}$$

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

$$9^{1}\times \frac{1}{12}x^{1}x^{3\left(-1\right)}$$

Multiply 3 times -1.

$$9^{1}\times \frac{1}{12}x^{1}x^{-3}$$

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

$$9^{1}\times \frac{1}{12}x^{1-3}$$

Add the exponents 1 and -3.

$$9^{1}\times \frac{1}{12}x^{-2}$$

Raise 9 to the power 1.

$$9\times \frac{1}{12}x^{-2}$$

Multiply 9 times \frac{1}{12}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{9^{1}x^{1}}{12^{1}x^{3}}$$

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

$$\frac{9^{1}x^{1-3}}{12^{1}}$$

Subtract 3 from 1.

$$\frac{9^{1}x^{-2}}{12^{1}}$$

Reduce the fraction \frac{9}{12} to lowest terms by extracting and canceling out 3.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9}{12}x^{1-3})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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