Solve 20x^4/4x^5

$$\frac{ 20 { x }^{ 4 } }{ 4 { x }^{ 5 } }$$

Evaluate

$$\frac{5}{x}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(20x^{4}\right)^{1}\times \frac{1}{4x^{5}}$$

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

$$20^{1}\left(x^{4}\right)^{1}\times \frac{1}{4}\times \frac{1}{x^{5}}$$

Use the Commutative Property of Multiplication.

$$20^{1}\times \frac{1}{4}\left(x^{4}\right)^{1}\times \frac{1}{x^{5}}$$

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

$$20^{1}\times \frac{1}{4}x^{4}x^{5\left(-1\right)}$$

Multiply 5 times -1.

$$20^{1}\times \frac{1}{4}x^{4}x^{-5}$$

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

$$20^{1}\times \frac{1}{4}x^{4-5}$$

Add the exponents 4 and -5.

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

Raise 20 to the power 1.

$$20\times \frac{1}{4}\times \frac{1}{x}$$

Multiply 20 times \frac{1}{4}.

$$5\times \frac{1}{x}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{20^{1}x^{4}}{4^{1}x^{5}}$$

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

$$\frac{20^{1}x^{4-5}}{4^{1}}$$

Subtract 5 from 4.

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

Divide 20 by 4.

$$5\times \frac{1}{x}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20}{4}x^{4-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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