Solve 5x^0/20x^4

$$\frac { 5 x ^ { 0 } } { 20 x ^ { 4 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(5x^{0}\right)^{1}\times \frac{1}{20x^{4}}$$

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

$$5^{1}\left(x^{0}\right)^{1}\times \frac{1}{20}\times \frac{1}{x^{4}}$$

Use the Commutative Property of Multiplication.

$$5^{1}\times \frac{1}{20}\left(x^{0}\right)^{1}\times \frac{1}{x^{4}}$$

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

$$5^{1}\times \frac{1}{20}x^{0}x^{4\left(-1\right)}$$

Multiply 4 times -1.

$$5^{1}\times \frac{1}{20}x^{0}x^{-4}$$

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

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

Raise 5 to the power 1.

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

Multiply 5 times \frac{1}{20}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{5^{1}x^{0}}{20^{1}x^{4}}$$

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

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

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

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

Cancel out 5 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{0}}{4x^{4}})$$

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

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

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

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}.

$$-\left(4x^{4}\right)^{-2}\times 4\times 4x^{4-1}$$

Simplify.

$$-16x^{3}\times \left(4x^{4}\right)^{-2}$$