Solve frac{120x^{2}y^3z^3}{30x^3yz^2}

$$\frac { 120 x ^ { 2 } y ^ { 3 } z ^ { 3 } } { 30 x ^ { 3 } y z ^ { 2 } }$$

Differentiate w.r.t. x

$$-4\times \left(\frac{y}{x}\right)^{2}z$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{120y^{3}z^{3}}{30yz^{2}}x^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4zy^{2}\times \frac{1}{x})$$

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}.

$$-4zy^{2}x^{-1-1}$$

Do the arithmetic.

$$\left(-4zy^{2}\right)x^{-2}$$

Evaluate

$$\frac{4zy^{2}}{x}$$

Solution Steps

Cancel out 30yx^{2}z^{2} in both numerator and denominator.

$$\frac{4zy^{2}}{x}$$