Solve (x^2)^4/x^20=

$$\frac { ( x ^ { 2 } ) ^ { 4 } } { x ^ { 20 } } =$$

Evaluate

$$\frac{1}{x^{12}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(x^{2}\right)^{4}\times \frac{1}{x^{20}}$$

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

$$x^{2\times 4}x^{20\left(-1\right)}$$

Multiply 2 times 4.

$$x^{8}x^{20\left(-1\right)}$$

Multiply 20 times -1.

$$x^{8}x^{-20}$$

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

$$x^{8-20}$$

Add the exponents 8 and -20.

$$x^{-12}$$

Differentiate w.r.t. x

$$-\frac{12}{x^{13}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{20}})$$

Rewrite x^{20} as x^{8}x^{12}. Cancel out x^{8} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{12}})$$

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).

$$-\left(x^{12}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{12})$$

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}.

$$-\left(x^{12}\right)^{-2}\times 12x^{12-1}$$

Simplify.

$$-12x^{11}\left(x^{12}\right)^{-2}$$