Solve 3/7x^{3}y^{8}z^8:(-1/4x^5y^6z^6)

$$\frac { 3 } { 7 } x ^ { 3 } y ^ { 8 } z ^ { 8 } : ( - \frac { 1 } { 4 } x ^ { 5 } y ^ { 6 } z ^ { 6 } )$$

Differentiate w.r.t. x

$$\frac{24\left(yz\right)^{2}}{7x^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y^{8}z^{8}}{7\left(-\frac{y^{6}z^{6}}{4}\right)}x^{3-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{12y^{2}z^{2}}{7}\right)x^{-2})$$

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}.

$$-2\left(-\frac{12y^{2}z^{2}}{7}\right)x^{-2-1}$$

Do the arithmetic.

$$\frac{24y^{2}z^{2}}{7}x^{-3}$$

Evaluate

$$-\frac{12\times \left(\frac{yz}{x}\right)^{2}}{7}$$

Solution Steps

Cancel out x^{3}y^{6}z^{6} in both numerator and denominator.

$$\frac{\frac{3}{7}y^{2}z^{2}}{-\frac{1}{4}x^{2}}$$