Solve x^frac{1{2}}x^-frac{1{6}}x^-frac{5{2}}

$$x ^ { \frac { 1 } { 2 } } x ^ { - \frac { 1 } { 6 } } x ^ { - \frac { 5 } { 2 } }$$

Evaluate

$$\frac{1}{x^{\frac{13}{6}}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\sqrt{x}x^{-\frac{1}{6}}x^{-\frac{5}{2}}$$

Use the Multiplication Rule for Exponents.

$$x^{\frac{1}{2}-\frac{1}{6}-\frac{5}{2}}$$

Add the exponents \frac{1}{2} and -\frac{1}{6}.

$$x^{\frac{1}{3}-\frac{5}{2}}$$

Add the exponents \frac{1}{3} and -\frac{5}{2}.

$$x^{-\frac{13}{6}}$$

Differentiate w.r.t. x

$$-\frac{13}{6x^{\frac{19}{6}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and -\frac{1}{6} to get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{1}{3}}x^{-\frac{5}{2}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and -\frac{5}{2} to get -\frac{13}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-\frac{13}{6}})$$

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

$$-\frac{13}{6}x^{-\frac{13}{6}-1}$$

Subtract 1 from -\frac{13}{6}.

$$-\frac{13}{6}x^{-\frac{19}{6}}$$