Solve (c^{2/3}+3c^{1/6})^2-3c^frac{1{12}}(3c^frac{1{4}}+2c^frac{3{4}})

$$( c ^ { \frac { 2 } { 3 } } + 3 c ^ { \frac { 1 } { 6 } } ) ^ { 2 } - 3 c ^ { \frac { 1 } { 12 } } ( 3 c ^ { \frac { 1 } { 4 } } + 2 c ^ { \frac { 3 } { 4 } } )$$

Differentiate w.r.t. c

$$\frac{4\sqrt[3]{c}}{3}$$

Steps Using Definition of a Derivative

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(c^{\frac{2}{3}}+3c^{\frac{1}{6}}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\left(c^{\frac{2}{3}}\right)^{2}+6c^{\frac{2}{3}}c^{\frac{1}{6}}+9\left(c^{\frac{1}{6}}\right)^{2}-3c^{\frac{1}{12}}\left(3c^{\frac{1}{4}}+2c^{\frac{3}{4}}\right))$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 2 to get \frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{4}{3}}+6c^{\frac{2}{3}}c^{\frac{1}{6}}+9\left(c^{\frac{1}{6}}\right)^{2}-3c^{\frac{1}{12}}\left(3c^{\frac{1}{4}}+2c^{\frac{3}{4}}\right))$$

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{1}{6} to get \frac{5}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{4}{3}}+6c^{\frac{5}{6}}+9\left(c^{\frac{1}{6}}\right)^{2}-3c^{\frac{1}{12}}\left(3c^{\frac{1}{4}}+2c^{\frac{3}{4}}\right))$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{6} and 2 to get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{4}{3}}+6c^{\frac{5}{6}}+9c^{\frac{1}{3}}-3c^{\frac{1}{12}}\left(3c^{\frac{1}{4}}+2c^{\frac{3}{4}}\right))$$

Use the distributive property to multiply -3c^{\frac{1}{12}} by 3c^{\frac{1}{4}}+2c^{\frac{3}{4}}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{4}{3}}+6c^{\frac{5}{6}}+9c^{\frac{1}{3}}-9c^{\frac{1}{3}}-6c^{\frac{5}{6}})$$

Combine 9c^{\frac{1}{3}} and -9c^{\frac{1}{3}} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{4}{3}}+6c^{\frac{5}{6}}-6c^{\frac{5}{6}})$$

Combine 6c^{\frac{5}{6}} and -6c^{\frac{5}{6}} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{4}{3}})$$

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

$$\frac{4}{3}c^{\frac{4}{3}-1}$$

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

$$\frac{4}{3}\sqrt[3]{c}$$

Evaluate

$$c^{\frac{4}{3}}$$