Solve x_{1}(sqrt{1/3}+sqrt{27})*sqrt{3}

$$x _ { 1 } ( \sqrt { \frac { 1 } { 3 } } + \sqrt { 27 } ) \times \sqrt { 3 }$$

Evaluate

$$10x_{1}$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$x_{1}\left(\frac{\sqrt{1}}{\sqrt{3}}+\sqrt{27}\right)\sqrt{3}$$

Calculate the square root of 1 and get 1.

$$x_{1}\left(\frac{1}{\sqrt{3}}+\sqrt{27}\right)\sqrt{3}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$x_{1}\left(\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{27}\right)\sqrt{3}$$

The square of \sqrt{3} is 3.

$$x_{1}\left(\frac{\sqrt{3}}{3}+\sqrt{27}\right)\sqrt{3}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$x_{1}\left(\frac{\sqrt{3}}{3}+3\sqrt{3}\right)\sqrt{3}$$

Combine \frac{\sqrt{3}}{3} and 3\sqrt{3} to get \frac{10}{3}\sqrt{3}.

$$x_{1}\times \frac{10}{3}\sqrt{3}\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$x_{1}\times \frac{10}{3}\times 3$$

Cancel out 3 and 3.

$$x_{1}\times 10$$

Differentiate w.r.t. x_1

$$10$$

Steps Using Definition of a Derivative

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\left(\frac{\sqrt{1}}{\sqrt{3}}+\sqrt{27}\right)\sqrt{3})$$

Calculate the square root of 1 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\left(\frac{1}{\sqrt{3}}+\sqrt{27}\right)\sqrt{3})$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\left(\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{27}\right)\sqrt{3})$$

The square of \sqrt{3} is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\left(\frac{\sqrt{3}}{3}+\sqrt{27}\right)\sqrt{3})$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\left(\frac{\sqrt{3}}{3}+3\sqrt{3}\right)\sqrt{3})$$

Combine \frac{\sqrt{3}}{3} and 3\sqrt{3} to get \frac{10}{3}\sqrt{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\times \frac{10}{3}\sqrt{3}\sqrt{3})$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\times \frac{10}{3}\times 3)$$

Cancel out 3 and 3.

$$\frac{\mathrm{d}}{\mathrm{d}x_{1}}(x_{1}\times 10)$$

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

$$10x_{1}^{1-1}$$

Subtract 1 from 1.

$$10x_{1}^{0}$$

For any term t except 0, t^{0}=1.

$$10\times 1$$

For any term t, t\times 1=t and 1t=t.

$$10$$