Solve [1/x]^3*[1/x]^3

$$[ \frac { 1 } { x } ] ^ { 3 } \cdot [ \frac { 1 } { x } ] ^ { 3 }$$

Evaluate

$$\frac{1}{x^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(\frac{1}{x}\right)^{3}\times \left(\frac{1}{x}\right)^{3}$$

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

$$x^{-3}x^{-3}$$

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

$$x^{-3-3}$$

Add the exponents -3 and -3.

$$x^{-6}$$

Differentiate w.r.t. x

$$-\frac{6}{x^{7}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{1}{x}\right)^{6})$$

To raise \frac{1}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{6}}{x^{6}})$$

Calculate 1 to the power of 6 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{6}})$$

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).

$$-\left(x^{6}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{6})$$

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}.

$$-\left(x^{6}\right)^{-2}\times 6x^{6-1}$$

Simplify.

$$-6x^{5}\left(x^{6}\right)^{-2}$$