Solve x^8y^12/x^16y^12

$$\frac { x ^ { 8 } y ^ { 12 } } { x ^ { 16 } y ^ { 12 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{x^{8}y^{12}}{x^{16}y^{12}}$$

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

$$x^{8-16}y^{12-12}$$

Subtract 16 from 8.

$$x^{-8}y^{12-12}$$

Subtract 12 from 12.

$$x^{-8}y^{0}$$

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

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

Cancel out x^{8}y^{12} in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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