Solve frac{x^3*x^frac{1{2}}}{x^-frac{1{3}}*x^-frac{1{6}}}

$$\frac { x ^ { 3 } \times x ^ { \frac { 1 } { 2 } } } { x ^ { - \frac { 1 } { 3 } } \times x ^ { - \frac { 1 } { 6 } } }$$

Evaluate

$$x^{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and \frac{1}{2} to get \frac{7}{2}.

$$\frac{x^{\frac{7}{2}}}{x^{-\frac{1}{3}}x^{-\frac{1}{6}}}$$

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

$$\frac{x^{\frac{7}{2}}}{x^{-\frac{1}{2}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{1}{2} from \frac{7}{2} to get 4.

$$x^{4}$$

Differentiate w.r.t. x

$$4x^{3}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and \frac{1}{2} to get \frac{7}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{7}{2}}}{x^{-\frac{1}{3}}x^{-\frac{1}{6}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{7}{2}}}{x^{-\frac{1}{2}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{1}{2} from \frac{7}{2} to get 4.

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

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

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

Subtract 1 from 4.

$$4x^{3}$$