Solve frac{a^{2}b^{2}c^8}{a^4b^6c^-2}

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

Evaluate

$$\frac{c^{10}}{a^{2}b^{4}}$$

Solution Steps

Cancel out a^{2}b^{2} in both numerator and denominator.

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

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

$$\frac{c^{10}}{a^{2}b^{4}}$$

Differentiate w.r.t. a

$$-\frac{2c^{10}}{a^{3}b^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{b^{2}c^{8}c^{2}}{b^{6}}a^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{c^{10}}{b^{4}}a^{-2})$$

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}.

$$-2\times \frac{c^{10}}{b^{4}}a^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{2c^{10}}{b^{4}}\right)a^{-3}$$