Solve a^frac{2{3}}(a^frac{7{3}}+2m^frac{1{3}})

$$a ^ { \frac { 2 } { 3 } } ( a ^ { \frac { 7 } { 3 } } + 2 m ^ { \frac { 1 } { 3 } } )$$

Differentiate w.r.t. a

$$3a^{2}+\frac{4\sqrt[3]{\frac{m}{a}}}{3}$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply a^{\frac{2}{3}} by a^{\frac{7}{3}}+2m^{\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{2}{3}}a^{\frac{7}{3}}+2a^{\frac{2}{3}}m^{\frac{1}{3}})$$

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{7}{3} to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{3}+2a^{\frac{2}{3}}m^{\frac{1}{3}})$$

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}.

$$3a^{3-1}+\frac{2}{3}\times 2\sqrt[3]{m}a^{\frac{2}{3}-1}$$

Subtract 1 from 3.

$$3a^{2}+\frac{2}{3}\times 2\sqrt[3]{m}a^{\frac{2}{3}-1}$$

Multiply \frac{2}{3} times 2m^{\frac{1}{3}}.

$$3a^{2}+\frac{4\sqrt[3]{m}}{3}a^{\frac{2}{3}-1}$$

Subtract 1 from \frac{2}{3}.

$$3a^{2}+\frac{4\sqrt[3]{m}}{3}a^{-\frac{1}{3}}$$

Evaluate

$$a^{3}+2\sqrt[3]{m}a^{\frac{2}{3}}$$