Solve x^3/6x^8

$$\frac { x ^ { 3 } } { 6 x ^ { 8 } }$$

Evaluate

$$\frac{1}{6x^{5}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(x^{1}\right)^{3}\times \frac{1}{6x^{8}}$$

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

$$1^{3}\left(x^{1}\right)^{3}\times \frac{1}{6}\times \frac{1}{x^{8}}$$

Use the Commutative Property of Multiplication.

$$1^{3}\times \frac{1}{6}\left(x^{1}\right)^{3}\times \frac{1}{x^{8}}$$

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

$$1^{3}\times \frac{1}{6}x^{3}x^{8\left(-1\right)}$$

Multiply 8 times -1.

$$1^{3}\times \frac{1}{6}x^{3}x^{-8}$$

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

$$1^{3}\times \frac{1}{6}x^{3-8}$$

Add the exponents 3 and -8.

$$1^{3}\times \frac{1}{6}x^{-5}$$

Raise 6 to the power -1.

$$\frac{1}{6}x^{-5}$$

Differentiate w.r.t. x

$$-\frac{5}{6x^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{6}x^{3-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{6}x^{-5})$$

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}.

$$-5\times \frac{1}{6}x^{-5-1}$$

Do the arithmetic.

$$-\frac{5}{6}x^{-6}$$