Solve frac{6x^{2}y^{4/3}z^-1/5}{5x^4y^3z^7}

$$\frac { 6 x ^ { 2 } y ^ { 4 / 3 } z ^ { - 1 / 5 } } { 5 x ^ { 4 } y ^ { 3 } z ^ { 7 } }$$

Evaluate

$$\frac{6}{5y^{\frac{5}{3}}x^{2}z^{\frac{36}{5}}}$$

Solution Steps

Cancel out y^{\frac{4}{3}}x^{2} in both numerator and denominator.

$$\frac{6z^{-\frac{1}{5}}}{5y^{\frac{5}{3}}x^{2}z^{7}}$$

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

$$\frac{6}{5y^{\frac{5}{3}}x^{2}z^{\frac{36}{5}}}$$

Differentiate w.r.t. x

$$-\frac{12}{5y^{\frac{5}{3}}x^{3}z^{\frac{36}{5}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{\frac{4}{3}}}{\sqrt[5]{z}\times 5y^{3}z^{7}}x^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6}{5y^{\frac{5}{3}}z^{\frac{36}{5}}}x^{-2})$$

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}.

$$-2\times \frac{6}{5y^{\frac{5}{3}}z^{\frac{36}{5}}}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{12}{5y^{\frac{5}{3}}z^{\frac{36}{5}}}\right)x^{-3}$$