Solve 2x^3yz/8x^-2y^4z^-2

$$\frac { 2 x ^ { 3 } y z } { 8 x ^ { - 2 } y ^ { 4 } z ^ { - 2 } }$$

Evaluate

$$\frac{\left(\frac{z}{y}\right)^{3}x^{5}}{4}$$

Solution Steps

Cancel out 2y in both numerator and denominator.

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

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

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

Differentiate w.r.t. x

$$\frac{5\times \left(\frac{z}{y}\right)^{3}x^{4}}{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2yzz^{2}}{8y^{4}}x^{3-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{z^{3}}{4y^{3}}x^{5})$$

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}.

$$5\times \frac{z^{3}}{4y^{3}}x^{5-1}$$

Do the arithmetic.

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