Solve frac{8a^{-2}b^{7}c^-6}{2a^8b^-2c^8}

$$\frac { 8 a ^ { - 2 } b ^ { 7 } c ^ { - 6 } } { 2 a ^ { 8 } b ^ { - 2 } c ^ { 8 } }$$

Evaluate

$$\frac{4b^{9}}{a^{10}c^{14}}$$

Solution Steps

Cancel out 2 in both numerator and denominator.

$$\frac{4c^{-6}a^{-2}b^{7}}{b^{-2}a^{8}c^{8}}$$

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

$$\frac{4c^{-6}a^{-2}b^{9}}{a^{8}c^{8}}$$

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

$$\frac{4b^{9}}{a^{10}c^{14}}$$

Differentiate w.r.t. a

$$-\frac{40b^{9}}{a^{11}c^{14}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{8b^{7}}{c^{6}\times \frac{2c^{8}}{b^{2}}}a^{-2-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{4b^{9}}{c^{14}}a^{-10})$$

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}.

$$-10\times \frac{4b^{9}}{c^{14}}a^{-10-1}$$

Do the arithmetic.

$$\left(-\frac{40b^{9}}{c^{14}}\right)a^{-11}$$