Solve 20x/25x^5

$$\frac { 20 x } { 25 x ^ { 5 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(20x^{1}\right)^{1}\times \frac{1}{25x^{5}}$$

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

$$20^{1}\left(x^{1}\right)^{1}\times \frac{1}{25}\times \frac{1}{x^{5}}$$

Use the Commutative Property of Multiplication.

$$20^{1}\times \frac{1}{25}\left(x^{1}\right)^{1}\times \frac{1}{x^{5}}$$

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

$$20^{1}\times \frac{1}{25}x^{1}x^{5\left(-1\right)}$$

Multiply 5 times -1.

$$20^{1}\times \frac{1}{25}x^{1}x^{-5}$$

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

$$20^{1}\times \frac{1}{25}x^{1-5}$$

Add the exponents 1 and -5.

$$20^{1}\times \frac{1}{25}x^{-4}$$

Raise 20 to the power 1.

$$20\times \frac{1}{25}x^{-4}$$

Multiply 20 times \frac{1}{25}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{20^{1}x^{1}}{25^{1}x^{5}}$$

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

$$\frac{20^{1}x^{1-5}}{25^{1}}$$

Subtract 5 from 1.

$$\frac{20^{1}x^{-4}}{25^{1}}$$

Reduce the fraction \frac{20}{25} to lowest terms by extracting and canceling out 5.

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

Differentiate w.r.t. x

$$-\frac{16}{5x^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20}{25}x^{1-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{5}x^{-4})$$

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}.

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

Do the arithmetic.

$$-\frac{16}{5}x^{-5}$$