derivative of sqrt(y/2) — step-by-step solution

Let's differentiate:

\frac{d}{d y} (\sqrt{\frac{y}{2}})

We differentiate using:

\frac{d}{d y} (\sqrt{f}) = \frac{d}{d y} (f) \cdot \frac{1}{2 \sqrt{f}}

(where f =

\frac{y}{2}

)

ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

\frac{d}{d y} (\sqrt{\frac{y}{2}})

\frac{d}{d y} (\frac{y}{2}) \cdot \frac{1}{2 \cdot \sqrt{\frac{y}{2}}}

\frac{d}{d y} (\frac{y}{2}) \cdot \frac{1}{2 \cdot \sqrt{\frac{y}{2}}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d y} (\frac{y}{2}) \cdot \frac{1}{2 \cdot \sqrt{\frac{y}{2}}}

\frac{\frac{d}{d y} (\frac{y}{2}) \cdot 1}{2 \cdot \sqrt{\frac{y}{2}}}

\frac{\frac{d}{d y} (\frac{y}{2})}{2 \cdot \sqrt{\frac{y}{2}}}

■A constant factor stays in front of the derivative, (k*u)' = k*u'. The derivative of x is 1:

\frac{d}{d y} (\frac{y}{2})

\frac{1}{2}

\frac{\frac{1}{2}}{2 \cdot \sqrt{\frac{y}{2}}}

ⓘ Dividing is multiplying by the reciprocal:

\frac{\frac{1}{2}}{2 \cdot \sqrt{\frac{y}{2}}}

\frac{1}{2} \cdot \sqrt{\frac{2}{y}} \cdot \frac{1}{2}

\frac{1}{2} \cdot \sqrt{\frac{2}{y}} \cdot \frac{1}{2}

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{1}{2} \cdot \frac{1}{2}

\frac{1}{4}

\frac{1}{4} \cdot \sqrt{\frac{2}{y}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{4} \cdot \sqrt{\frac{2}{y}}

\frac{1 \cdot \sqrt{\frac{2}{y}}}{4}

\frac{\sqrt{\frac{2}{y}}}{4}

▶Derivative:

\frac{\sqrt{\frac{2}{y}}}{4}