Solve frac{18a^{3}b^-2*5ab^2}{9a^4b*(-ab^3)}

$$\frac { 18 a ^ { 3 } b ^ { - 2 } \cdot 5 a b ^ { 2 } } { 9 a ^ { 4 } b \cdot ( - a b ^ { 3 } ) }$$

Differentiate w.r.t. a

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{90ab^{2}}{b^{2}\left(-9ab^{4}\right)}a^{3-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-\frac{10}{b^{4}}\right)\times \frac{1}{a})$$

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}.

$$-\left(-\frac{10}{b^{4}}\right)a^{-1-1}$$

Do the arithmetic.

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

Evaluate

$$-\frac{10}{ab^{4}}$$