Solve {x}^-frac{1{2}}+2{x}^-frac{1{2}}+{x}^frac{1{2}}

$${ x }^{ - \frac{ 1 }{ 2 } } +2 { x }^{ - \frac{ 1 }{ 2 } } + { x }^{ \frac{ 1 }{ 2 } }$$

Differentiate w.r.t. x

$$\frac{x-3}{2x^{\frac{3}{2}}}$$

Steps Using Derivative Rule for Sum

Combine x^{-\frac{1}{2}} and 2x^{-\frac{1}{2}} to get 3x^{-\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{-\frac{1}{2}}+x^{\frac{1}{2}})$$

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}.

$$-\frac{1}{2}\times 3x^{-\frac{1}{2}-1}+\frac{1}{2}x^{\frac{1}{2}-1}$$

Multiply -\frac{1}{2} times 3.

$$-\frac{3}{2}x^{-\frac{1}{2}-1}+\frac{1}{2}x^{\frac{1}{2}-1}$$

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

$$-\frac{3}{2}x^{-\frac{3}{2}}+\frac{1}{2}x^{\frac{1}{2}-1}$$

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

$$-\frac{3}{2}x^{-\frac{3}{2}}+\frac{1}{2}x^{-\frac{1}{2}}$$

Evaluate

$$\frac{x+3}{\sqrt{x}}$$

Solution Steps

Combine x^{-\frac{1}{2}} and 2x^{-\frac{1}{2}} to get 3x^{-\frac{1}{2}}.

$$3x^{-\frac{1}{2}}+x^{\frac{1}{2}}$$