Solve g(x)=2x^5/x^8

$$g ( x ) = \frac { 2 x ^ { 5 } } { x ^ { 8 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(2x^{5}\right)^{1}\times \frac{1}{x^{8}}$$

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

$$2^{1}\left(x^{5}\right)^{1}\times \frac{1}{1}\times \frac{1}{x^{8}}$$

Use the Commutative Property of Multiplication.

$$2^{1}\times \frac{1}{1}\left(x^{5}\right)^{1}\times \frac{1}{x^{8}}$$

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

$$2^{1}\times \frac{1}{1}x^{5}x^{8\left(-1\right)}$$

Multiply 8 times -1.

$$2^{1}\times \frac{1}{1}x^{5}x^{-8}$$

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

$$2^{1}\times \frac{1}{1}x^{5-8}$$

Add the exponents 5 and -8.

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

Raise 2 to the power 1.

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

Differentiate w.r.t. x

$$-\frac{6}{x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{1}x^{5-8})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-6x^{-4}$$