Solve (3g^2/5-7g^1/6)-(g^2/5-3g^1/6)

$$( 3 g ^ { 2 / 5 } - 7 g ^ { 1 / 6 } ) - ( g ^ { 2 / 5 } - 3 g ^ { 1 / 6 } )$$

Evaluate

$$2g^{\frac{2}{5}}-4\sqrt[6]{g}$$

Solution Steps

To find the opposite of g^{\frac{2}{5}}-3g^{\frac{1}{6}}, find the opposite of each term.

$$3g^{\frac{2}{5}}-7g^{\frac{1}{6}}-g^{\frac{2}{5}}+3g^{\frac{1}{6}}$$

Combine 3g^{\frac{2}{5}} and -g^{\frac{2}{5}} to get 2g^{\frac{2}{5}}.

$$2g^{\frac{2}{5}}-7g^{\frac{1}{6}}+3g^{\frac{1}{6}}$$

Combine -7g^{\frac{1}{6}} and 3g^{\frac{1}{6}} to get -4g^{\frac{1}{6}}.

$$2g^{\frac{2}{5}}-4g^{\frac{1}{6}}$$

Differentiate w.r.t. g

$$\frac{4}{5g^{\frac{3}{5}}}-\frac{2}{3g^{\frac{5}{6}}}$$

Steps Using Derivative Rule for Sum

To find the opposite of g^{\frac{2}{5}}-3g^{\frac{1}{6}}, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}g}(3g^{\frac{2}{5}}-7g^{\frac{1}{6}}-g^{\frac{2}{5}}+3g^{\frac{1}{6}})$$

Combine 3g^{\frac{2}{5}} and -g^{\frac{2}{5}} to get 2g^{\frac{2}{5}}.

$$\frac{\mathrm{d}}{\mathrm{d}g}(2g^{\frac{2}{5}}-7g^{\frac{1}{6}}+3g^{\frac{1}{6}})$$

Combine -7g^{\frac{1}{6}} and 3g^{\frac{1}{6}} to get -4g^{\frac{1}{6}}.

$$\frac{\mathrm{d}}{\mathrm{d}g}(2g^{\frac{2}{5}}-4g^{\frac{1}{6}})$$

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}.

$$\frac{2}{5}\times 2g^{\frac{2}{5}-1}+\frac{1}{6}\left(-4\right)g^{\frac{1}{6}-1}$$

Multiply \frac{2}{5} times 2.

$$\frac{4}{5}g^{\frac{2}{5}-1}+\frac{1}{6}\left(-4\right)g^{\frac{1}{6}-1}$$

Subtract 1 from \frac{2}{5}.

$$\frac{4}{5}g^{-\frac{3}{5}}+\frac{1}{6}\left(-4\right)g^{\frac{1}{6}-1}$$

Multiply \frac{1}{6} times -4.

$$\frac{4}{5}g^{-\frac{3}{5}}-\frac{2}{3}g^{\frac{1}{6}-1}$$

Subtract 1 from \frac{1}{6}.

$$\frac{4}{5}g^{-\frac{3}{5}}-\frac{2}{3}g^{-\frac{5}{6}}$$