Solve 5x^4/x^5

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 5 times -1.

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

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

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

Add the exponents 4 and -5.

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

Raise 5 to the power 1.

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

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.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{1}x^{4-5})$$

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.

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