Solve x^{frac{1/3}*x^frac{1{2}}*x^frac{1{1}}}{x^frac{1{6}}*x^frac{1{12}}}

$$\frac { x ^ { \frac { 1 } { 3 } } \cdot x ^ { \frac { 1 } { 2 } } \cdot x ^ { \frac { 1 } { 1 } } } { x ^ { \frac { 1 } { 6 } } \cdot x ^ { \frac { 1 } { 12 } } }$$

Differentiate w.r.t. x

$$\frac{19x^{\frac{7}{12}}}{12}$$

Steps Using Definition of a Derivative

Divide 1 by 1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{1}{3}}x^{\frac{1}{2}}x^{1}}{x^{\frac{1}{6}}x^{\frac{1}{12}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{5}{6}}x^{1}}{x^{\frac{1}{6}}x^{\frac{1}{12}}})$$

To multiply powers of the same base, add their exponents. Add \frac{5}{6} and 1 to get \frac{11}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{11}{6}}}{x^{\frac{1}{6}}x^{\frac{1}{12}}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{6} and \frac{1}{12} to get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{11}{6}}}{x^{\frac{1}{4}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{1}{4} from \frac{11}{6} to get \frac{19}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{19}{12}})$$

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

$$\frac{19}{12}x^{\frac{19}{12}-1}$$

Subtract 1 from \frac{19}{12}.

$$\frac{19}{12}x^{\frac{7}{12}}$$

Evaluate

$$x^{\frac{19}{12}}$$