Question

Find from first principle the derivative of f(x)=root of X with respect to x

Answer 1

Given:

`f(x)=√(x)`

Required:

To find the derivative of the given function by using the first principle.

Step-by-step explanation:

To find the derivative by the first principle we will use the limit method.

Rationalise the denominator

Use the formula:

`(a+b)(a-b)=a^2-b^2`
`\begin{gathered} f^(\prime)(x)=\lim_(h\to0)(x+h-x)/(h(√(x+h)+√(x))) \\ f^(\prime)(x)=\lim_(h\to0)(h)/(h(√(x+h)+√(x))) \\ f^(\prime)(x)=\lim_(h\to0)(1)/((√(x+h)+√(x))) \end{gathered}`

Now apply the given limit

`\begin{gathered} f^(\prime)(x)=(1)/((√(x+0)-√(x))) \\ f^(\prime)(x)=(1)/(2√(x)) \end{gathered}`

Final answer:

Thus the derivative of the given function is

`f^(\prime)(x)=(1)/(2√(x))`