Solve (x^9)^-2/x^4

$$\frac { ( x ^ { 9 } ) ^ { - 2 } } { x ^ { 4 } }$$

Evaluate

$$\frac{1}{x^{22}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(x^{9}\right)^{-2}\times \frac{1}{x^{4}}$$

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

$$x^{9\left(-2\right)}x^{4\left(-1\right)}$$

Multiply 9 times -2.

$$x^{-18}x^{4\left(-1\right)}$$

Multiply 4 times -1.

$$x^{-18}x^{-4}$$

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

$$x^{-18-4}$$

Add the exponents -18 and -4.

$$x^{-22}$$

Differentiate w.r.t. x

$$-\frac{22}{x^{23}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 9 and -2 to get -18.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-18}}{x^{4}})$$

Rewrite x^{4} as x^{-18}x^{22}. Cancel out x^{-18} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{22}})$$

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).

$$-\left(x^{22}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{22})$$

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}.

$$-\left(x^{22}\right)^{-2}\times 22x^{22-1}$$

Simplify.

$$-22x^{21}\left(x^{22}\right)^{-2}$$