Solve 5xy^4z^2/20x^3yz^6

$$\frac { 5 x y ^ { 4 } z ^ { 2 } } { 20 x ^ { 3 } y z ^ { 6 } }$$

Differentiate w.r.t. x

$$-\frac{\left(\frac{y}{x}\right)^{3}}{2z^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5z^{2}y^{4}}{20yz^{6}}x^{1-3})$$

Do the arithmetic.

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

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}.

$$-2\times \frac{y^{3}}{4z^{4}}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{y^{3}}{2z^{4}}\right)x^{-3}$$

Evaluate

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

Solution Steps

Cancel out 5xyz^{2} in both numerator and denominator.

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