Question

Find the second and third derivative of
`y = √(x)`

Answer 1

`\begin{gathered} \text{The function }y=\sqrt[]{x},\text{ can be expressed as:} \\ y=x^{(1)/(2)} \end{gathered}`

We can use the power rule to get the second and third derivative of the function.

`\begin{gathered} \text{First derivative:} \\ y^(\prime)=\mleft((1)/(2)\mright)x^{(1)/(2)-1} \\ y^(\prime)=((1)/(2))x^{-(1)/(2)} \\ y^(\prime)=\frac{x^{-(1)/(2)}}{2} \end{gathered}`
`\begin{gathered} \text{Second derivative} \\ y^(\prime)^(\prime)=(-(1)/(2))\frac{x^{-(1)/(2)-1}}{2} \\ y^(\prime\prime)=-\frac{x^{-(3)/(2)}}{4}\text{ or }y^(\prime\prime)=-\frac{1}{4x^{(3)/(2)}} \\ \end{gathered}`
`\begin{gathered} \text{Third derivative} \\ y^(\prime)^(\prime)^(\prime)=(-(3)/(2))-\frac{x^{-(3)/(2)-1}}{4} \\ y^(\prime\prime\prime)=\frac{3x^{-(5)/(2)}}{8}\text{ or }y^(\prime\prime\prime)=\frac{3}{8x^{(5)/(2)}} \end{gathered}`