Solve 4/9^3a^-frac{3{2}}a^frac{1{6}}

$$\frac { 4 } { 9 ^ { 3 } } a ^ { - \frac { 3 } { 2 } } a ^ { \frac { 1 } { 6 } }$$

Evaluate

$$\frac{4}{729a^{\frac{4}{3}}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -\frac{3}{2} and \frac{1}{6} to get -\frac{4}{3}.

$$\frac{4}{9^{3}}a^{-\frac{4}{3}}$$

Calculate 9 to the power of 3 and get 729.

$$\frac{4}{729}a^{-\frac{4}{3}}$$

Differentiate w.r.t. a

$$-\frac{16}{2187a^{\frac{7}{3}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -\frac{3}{2} and \frac{1}{6} to get -\frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{4}{9^{3}}a^{-\frac{4}{3}})$$

Calculate 9 to the power of 3 and get 729.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{4}{729}a^{-\frac{4}{3}})$$

The derivative of ax^{n} is nax^{n-1}.

$$-\frac{4}{3}\times \frac{4}{729}a^{-\frac{4}{3}-1}$$

Multiply -\frac{4}{3} times \frac{4}{729} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-\frac{16}{2187}a^{-\frac{4}{3}-1}$$

Subtract 1 from -\frac{4}{3}.

$$-\frac{16}{2187}a^{-\frac{7}{3}}$$