Solve frac{12{x}^frac{-1{6}}sqrt{4{y}^frac{2{3}}x^-2}}{3y^0{x}^frac{1{2}}sqrt[3]{-64{y}^frac{-1{2}}x}}

$$\frac{ 12 { x }^{ \frac{ -1 }{ 6 } } \sqrt{ 4 { y }^{ \frac{ 2 }{ 3 } } { x }^{ -2 } } }{ 3 { y }^{ 0 } { x }^{ \frac{ 1 }{ 2 } } \sqrt[3]{ -64 { y }^{ \frac{ -1 }{ 2 } } x } }$$

Differentiate w.r.t. x

$$\frac{4\sqrt{y}}{x^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12\sqrt{\frac{4y^{\frac{2}{3}}}{x^{2}}}}{3\sqrt[3]{-\frac{64x}{\sqrt{y}}}y^{0}}x^{-\frac{1}{6}-\frac{1}{2}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{2\sqrt{y}}{\sqrt[3]{x}|x|}\right)x^{-\frac{2}{3}})$$

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}.

$$-\frac{2}{3}\left(-\frac{2\sqrt{y}}{\sqrt[3]{x}|x|}\right)x^{-\frac{2}{3}-1}$$

Do the arithmetic.

$$\frac{4\sqrt{y}}{3\sqrt[3]{x}|x|}x^{-\frac{5}{3}}$$

Evaluate

$$-\frac{2\sqrt{y}}{x^{2}}$$