Solve frac{12x^{-4}y^{-6}z^{2}b^{-2}}{28z^2x^2y^-7b^14}

$$\frac { 12 x ^ { - 4 } y ^ { - 6 } z ^ { 2 } b ^ { - 2 } } { 28 z ^ { 2 } x ^ { 2 } y ^ { - 7 } b ^ { 14 } }$$

Evaluate

$$\frac{3y}{7x^{6}b^{16}}$$

Solution Steps

Cancel out 4z^{2} in both numerator and denominator.

$$\frac{3y^{-6}x^{-4}b^{-2}}{7y^{-7}x^{2}b^{14}}$$

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

$$\frac{3x^{-4}b^{-2}y^{1}}{7x^{2}b^{14}}$$

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

$$\frac{3y^{1}}{7x^{6}b^{16}}$$

Calculate y to the power of 1 and get y.

$$\frac{3y}{7x^{6}b^{16}}$$

Differentiate w.r.t. x

$$-\frac{18y}{7x^{7}b^{16}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12z^{2}}{b^{2}y^{6}\times \frac{28z^{2}b^{14}}{y^{7}}}x^{-4-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y}{7b^{16}}x^{-6})$$

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}.

$$-6\times \frac{3y}{7b^{16}}x^{-6-1}$$

Do the arithmetic.

$$\left(-\frac{18y}{7b^{16}}\right)x^{-7}$$