Solve frac{18x^{5}y^2z^3}{3x^7y^2z}

$$\frac { 18 x ^ { 5 } y ^ { 2 } z ^ { 3 } } { 3 x ^ { 7 } y ^ { 2 } z }$$

Differentiate w.r.t. x

$$-\frac{12z^{2}}{x^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{18y^{2}z^{3}}{3zy^{2}}x^{5-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6z^{2}x^{-2})$$

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}.

$$-2\times 6z^{2}x^{-2-1}$$

Do the arithmetic.

$$\left(-12z^{2}\right)x^{-3}$$

Evaluate

$$6\times \left(\frac{z}{x}\right)^{2}$$

Solution Steps

Cancel out 3zy^{2}x^{5} in both numerator and denominator.

$$\frac{6z^{2}}{x^{2}}$$