Solve 5x^-2/5x^2

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

Evaluate

$$\frac{1}{x^{4}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 2 times -1.

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

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

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

Add the exponents -2 and -2.

$$5^{1}\times \frac{1}{5}x^{-4}$$

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

$$5^{1-1}x^{-4}$$

Add the exponents 1 and -1.

$$5^{0}x^{-4}$$

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

$$1x^{-4}$$

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

$$x^{-4}$$

Differentiate w.r.t. x

$$-\frac{4}{x^{5}}$$

Solution Steps

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

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

Do the arithmetic.

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

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}.

$$-4x^{-4-1}$$

Do the arithmetic.

$$-4x^{-5}$$