Solve 2x^2/2x^5

$$\frac { 2 x ^ { 2 } } { 2 x ^ { 5 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 5 times -1.

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

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

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

Add the exponents 2 and -5.

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

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

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

Add the exponents 1 and -1.

$$2^{0}x^{-3}$$

For any term t except 0, t^{0}=1.

$$1x^{-3}$$

For any term t, t\times 1=t and 1t=t.

$$x^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{2^{1}x^{2}}{2^{1}x^{5}}$$

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

$$2^{1-1}x^{2-5}$$

Subtract 1 from 1.

$$2^{0}x^{2-5}$$

For any number a except 0, a^{0}=1.

$$x^{2-5}$$

Subtract 5 from 2.

$$x^{-3}$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

$$-3x^{-3-1}$$

Do the arithmetic.

$$-3x^{-4}$$