Solve frac{4{x}^frac{2{3}}*12{x}^frac{1{6}}}{3{x}^frac{5{12}}*6{x}^frac{1{3}}}

$$\frac{ 4 { x }^{ \frac{ 2 }{ 3 } } \times 12 { x }^{ \frac{ 1 }{ 6 } } }{ 3 { x }^{ \frac{ 5 }{ 12 } } \times 6 { x }^{ \frac{ 1 }{ 3 } } }$$

Differentiate w.r.t. x

$$\frac{2}{9x^{\frac{11}{12}}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{48\sqrt[6]{x}}{18\sqrt[3]{x}}x^{\frac{2}{3}-\frac{5}{12}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8}{3\sqrt[6]{x}}\sqrt[4]{x})$$

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}.

$$\frac{1}{4}\times \frac{8}{3\sqrt[6]{x}}x^{\frac{1}{4}-1}$$

Do the arithmetic.

$$\frac{2}{3\sqrt[6]{x}}x^{-\frac{3}{4}}$$

Evaluate

$$\frac{8\sqrt[12]{x}}{3}$$