Solve 5u^0/5u^8

$$\frac { 5 u ^ { 0 } } { 5 u ^ { 8 } }$$

Evaluate

$$\frac{1}{u^{8}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(5u^{0}\right)^{1}\times \frac{1}{5u^{8}}$$

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

$$5^{1}\left(u^{0}\right)^{1}\times \frac{1}{5}\times \frac{1}{u^{8}}$$

Use the Commutative Property of Multiplication.

$$5^{1}\times \frac{1}{5}\left(u^{0}\right)^{1}\times \frac{1}{u^{8}}$$

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

$$5^{1}\times \frac{1}{5}u^{0}u^{8\left(-1\right)}$$

Multiply 8 times -1.

$$5^{1}\times \frac{1}{5}u^{0}u^{-8}$$

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

$$5^{1}\times \frac{1}{5}u^{-8}$$

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

$$5^{1-1}u^{-8}$$

Add the exponents 1 and -1.

$$5^{0}u^{-8}$$

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

$$1u^{-8}$$

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

$$u^{-8}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{5^{1}u^{0}}{5^{1}u^{8}}$$

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

$$5^{1-1}u^{-8}$$

Subtract 1 from 1.

$$5^{0}u^{-8}$$

For any number a except 0, a^{0}=1.

$$u^{-8}$$

Differentiate w.r.t. u

$$-\frac{8}{u^{9}}$$

Steps Using Chain Rule

Cancel out 5 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}u}(\frac{u^{0}}{u^{8}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}u}(\frac{1}{u^{8}})$$

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).

$$-\left(u^{8}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}u}(u^{8})$$

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}.

$$-\left(u^{8}\right)^{-2}\times 8u^{8-1}$$

Simplify.

$$-8u^{7}\left(u^{8}\right)^{-2}$$