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

User MoSwilam
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1 Answer

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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.


\begin{gathered} f^(\prime)(x)=\lim_(h\to0)(f(x+h)-f(x))/(h) \\ f^(\prime)(x)=\operatorname{\lim}_(h\to0)(√(x+h)-√(x))/(h) \end{gathered}

Rationalise the denominator


f^(\prime)(x)=\operatorname{\lim}_(h\to0)(√(x+h)-√(x))/(h)*(√(x+h)+√(x))/(√(x+h)+√(x))

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))

User Fiil
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