Solve frac{a^{-5}b^{-2}c^-7}{a^3b^8c^6}

$$\frac { a ^ { - 5 } b ^ { - 2 } c ^ { - 7 } } { a ^ { 3 } b ^ { 8 } c ^ { 6 } }$$

Differentiate w.r.t. a

$$-\frac{8}{a^{9}b^{10}c^{13}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{b^{2}c^{7}c^{6}b^{8}}a^{-5-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{b^{10}c^{13}}a^{-8})$$

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}.

$$-8\times \frac{1}{b^{10}c^{13}}a^{-8-1}$$

Do the arithmetic.

$$\left(-\frac{8}{b^{10}c^{13}}\right)a^{-9}$$

Evaluate

$$\frac{1}{a^{8}b^{10}c^{13}}$$

Solution Steps

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

$$\frac{1}{a^{8}b^{10}c^{13}}$$