Solve frac{3x^{2}y^{-3}z^{4}}{yx^2z^3*2xy^4z^-3}

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

Evaluate

$$\frac{3z^{4}}{2xy^{8}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{3x^{2}y^{-3}z^{4}}{y^{5}x^{2}z^{3}\times 2xz^{-3}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{3x^{2}y^{-3}z^{4}}{y^{5}x^{3}z^{3}\times 2z^{-3}}$$

Multiply z^{3} and z^{-3} to get 1.

$$\frac{3x^{2}y^{-3}z^{4}}{y^{5}x^{3}\times 2}$$

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

$$\frac{3y^{-3}z^{4}}{2xy^{5}}$$

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

$$\frac{3z^{4}}{2xy^{8}}$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3z^{4}}{2xy^{8}}x^{0})$$

For any number a except 0, a^{0}=1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3z^{4}}{2xy^{8}})$$

The derivative of a constant term is 0.

$$0$$