Solve (frac{4x^{2}y^rz}{12x^4yz^-3})^prime

$$( \frac { 4 x ^ { 2 } y ^ { r } z } { 12 x ^ { 4 } y z ^ { - 3 } } ) ^ { \prime }$$

Evaluate

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4zy^{r}z^{3}}{12y}x^{2-4})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{2z^{4}y^{r-1}}{3}\right)x^{-3}$$

Differentiate w.r.t. x

$$2\times \left(\frac{z}{x}\right)^{4}y^{r-1}$$