Solve 2x^1div6*2x^-7div6

$$2 { x }^{ 1 \div 6 } \times 2 { x }^{ -7 \div 6 }$$

Evaluate

$$\frac{4}{x}$$

Solution Steps

Multiply 2 and 2 to get 4.

$$4x^{\frac{1}{6}}x^{\frac{-7}{6}}$$

Fraction \frac{-7}{6} can be rewritten as -\frac{7}{6} by extracting the negative sign.

$$4x^{\frac{1}{6}}x^{-\frac{7}{6}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{6} and -\frac{7}{6} to get -1.

$$4x^{-1}$$

Differentiate w.r.t. x

$$-\frac{4}{x^{2}}$$

Steps Using Definition of a Derivative

Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{\frac{1}{6}}x^{\frac{-7}{6}})$$

Fraction \frac{-7}{6} can be rewritten as -\frac{7}{6} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{\frac{1}{6}}x^{-\frac{7}{6}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{6} and -\frac{7}{6} to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{-1})$$

The derivative of ax^{n} is nax^{n-1}.

$$-4x^{-1-1}$$

Subtract 1 from -1.

$$-4x^{-2}$$