Solve frac{12xysqrt[3]{x^2y}}{6x^2y}

$$\frac{ 12xy \sqrt[3]{ { x }^{ 2 } y } }{ 6 { x }^{ 2 } y }$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2\sqrt[3]{y}x^{\frac{2}{3}}\times \frac{1}{x})$$

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}.

$$-2\sqrt[3]{y}x^{\frac{2}{3}}x^{-1-1}$$

Do the arithmetic.

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

Evaluate

$$2\sqrt[3]{\frac{y}{x}}$$