Solve 2a^3b^6/5a^9b^9

$$\frac { 2 a ^ { 3 } b ^ { 6 } } { 5 a ^ { 9 } b ^ { 9 } }$$

Evaluate

$$\frac{2}{5b^{3}a^{6}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{2^{1}a^{3}b^{6}}{5^{1}a^{9}b^{9}}$$

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

$$\frac{2^{1}}{5^{1}}a^{3-9}b^{6-9}$$

Subtract 9 from 3.

$$\frac{2^{1}}{5^{1}}a^{-6}b^{6-9}$$

Subtract 9 from 6.

$$\frac{2^{1}}{5^{1}}\times \frac{1}{a^{6}}b^{-3}$$

Divide 2 by 5.

$$\frac{2}{5}\times \frac{1}{a^{6}}\times \frac{1}{b^{3}}$$

Differentiate w.r.t. a

$$-\frac{12}{5b^{3}a^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{2b^{6}}{5b^{9}}a^{3-9})$$

Do the arithmetic.

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

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}.

$$-6\times \frac{2}{5b^{3}}a^{-6-1}$$

Do the arithmetic.

$$\left(-\frac{12}{5b^{3}}\right)a^{-7}$$