Solve frac{30x^{4}y^{8}z^9}{5ax^6y^2z^9}

$$\frac { 30 x ^ { 4 } y ^ { 8 } z ^ { 9 } } { 5 a x ^ { 6 } y ^ { 2 } z ^ { 9 } }$$

Differentiate w.r.t. x

$$-\frac{12y^{6}}{ax^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{30y^{8}z^{9}}{5ay^{2}z^{9}}x^{4-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{6}}{a}x^{-2})$$

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}.

$$-2\times \frac{6y^{6}}{a}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{12y^{6}}{a}\right)x^{-3}$$

Evaluate

$$\frac{6y^{6}}{ax^{2}}$$

Solution Steps

Cancel out 5y^{2}x^{4}z^{9} in both numerator and denominator.

$$\frac{6y^{6}}{ax^{2}}$$