Solve 6x^6/14x^9

$$\frac { 6 x ^ { 6 } } { 14 x ^ { 9 } }$$

Evaluate

$$\frac{3}{7x^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(6x^{6}\right)^{1}\times \frac{1}{14x^{9}}$$

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

$$6^{1}\left(x^{6}\right)^{1}\times \frac{1}{14}\times \frac{1}{x^{9}}$$

Use the Commutative Property of Multiplication.

$$6^{1}\times \frac{1}{14}\left(x^{6}\right)^{1}\times \frac{1}{x^{9}}$$

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

$$6^{1}\times \frac{1}{14}x^{6}x^{9\left(-1\right)}$$

Multiply 9 times -1.

$$6^{1}\times \frac{1}{14}x^{6}x^{-9}$$

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

$$6^{1}\times \frac{1}{14}x^{6-9}$$

Add the exponents 6 and -9.

$$6^{1}\times \frac{1}{14}x^{-3}$$

Raise 6 to the power 1.

$$6\times \frac{1}{14}x^{-3}$$

Multiply 6 times \frac{1}{14}.

$$\frac{3}{7}x^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{6^{1}x^{6}}{14^{1}x^{9}}$$

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

$$\frac{6^{1}x^{6-9}}{14^{1}}$$

Subtract 9 from 6.

$$\frac{6^{1}x^{-3}}{14^{1}}$$

Reduce the fraction \frac{6}{14} to lowest terms by extracting and canceling out 2.

$$\frac{3}{7}x^{-3}$$

Differentiate w.r.t. x

$$-\frac{9}{7x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6}{14}x^{6-9})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{7}x^{-3})$$

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}.

$$-3\times \frac{3}{7}x^{-3-1}$$

Do the arithmetic.

$$-\frac{9}{7}x^{-4}$$