Solve 3xyz/9x^2y^3z^4

$$\frac { 3 x y z } { 9 x ^ { 2 } y ^ { 3 } z ^ { 4 } }$$

Differentiate w.r.t. x

$$-\frac{1}{3z^{3}\left(xy\right)^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3yz}{9y^{3}z^{4}}x^{1-2})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{1}{3y^{2}z^{3}}\right)x^{-2}$$

Evaluate

$$\frac{1}{3xy^{2}z^{3}}$$

Solution Steps

Cancel out 3xyz in both numerator and denominator.

$$\frac{1}{3xy^{2}z^{3}}$$