Solve 6x^{frac{2/3}y^{4/3}z^-1/5}{5x^4y^frac{2{2}}z^2}

$$\frac { 6 x ^ { \frac { 2 } { 3 } } y ^ { 4 / 3 } z ^ { - 1 / 5 } } { 5 x ^ { 4 } y ^ { \frac { 2 } { 2 } } z ^ { 2 } }$$

Differentiate w.r.t. x

$$-\frac{4\sqrt[3]{y}}{z^{\frac{11}{5}}x^{\frac{13}{3}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{\frac{4}{3}}}{\sqrt[5]{z}\times 5y^{1}z^{2}}x^{\frac{2}{3}-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6\sqrt[3]{y}}{5z^{\frac{11}{5}}}x^{-\frac{10}{3}})$$

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}.

$$-\frac{10}{3}\times \frac{6\sqrt[3]{y}}{5z^{\frac{11}{5}}}x^{-\frac{10}{3}-1}$$

Do the arithmetic.

$$\left(-\frac{4\sqrt[3]{y}}{z^{\frac{11}{5}}}\right)x^{-\frac{13}{3}}$$

Evaluate

$$\frac{6\sqrt[3]{y}}{5z^{\frac{11}{5}}x^{\frac{10}{3}}}$$