Solve frac{x^{3}y^{5}z^-4}{v^3x^-2z^2}

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

Evaluate

$$\frac{\left(xy\right)^{5}}{v^{3}z^{6}}$$

Solution Steps

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

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

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

$$\frac{x^{5}y^{5}}{v^{3}z^{6}}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{5}}{z^{4}z^{2}v^{3}}x^{3-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{5}}{v^{3}z^{6}}x^{5})$$

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}.

$$5\times \frac{y^{5}}{v^{3}z^{6}}x^{5-1}$$

Do the arithmetic.

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