Solve frac{-120a^{8}b^{9}c^5}{10a^7b^7c^3}

$$\frac { - 120 a ^ { 8 } b ^ { 9 } c ^ { 5 } } { 10 a ^ { 7 } b ^ { 7 } c ^ { 3 } }$$

Differentiate w.r.t. a

$$-12\left(bc\right)^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-\frac{120c^{5}b^{9}}{10c^{3}b^{7}}\right)a^{8-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-12b^{2}c^{2}\right)a^{1})$$

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}.

$$\left(-12b^{2}c^{2}\right)a^{1-1}$$

Do the arithmetic.

$$\left(-12b^{2}c^{2}\right)a^{0}$$

For any term t except 0, t^{0}=1.

$$\left(-12b^{2}c^{2}\right)\times 1$$

For any term t, t\times 1=t and 1t=t.

$$-12b^{2}c^{2}$$

Evaluate

$$-12a\left(bc\right)^{2}$$

Solution Steps

Cancel out 10c^{3}a^{7}b^{7} in both numerator and denominator.

$$-12ab^{2}c^{2}$$