Derivative of square root using the power rule

Let $f(x) = \sqrt{x}$ .

Rewrite $f(x) = \sqrt{x}$ as $f(x) = x^{1/2}$ , use the power rule to find $f'(x)$ , and then evaluate $f'(4)$ . What is $f'(4)$ ?

Hint

Apply the power rule $\frac{d}{dx}[x^n] = n x^{n-1}$ with $n = \frac{1}{2}$ , then substitute $x = 4$ into the resulting expression before simplifying.

Step-by-step solution

Rewrite $f(x) = \sqrt{x}$ as $f(x) = x^{1/2}$ . Applying the power rule with $n = \frac{1}{2}$ gives $f'(x) = \frac{1}{2}x^{1/2 - 1} = \frac{1}{2}x^{-1/2}$ .

This can be rewritten as $f'(x) = \frac{1}{2\sqrt{x}}$ . Substituting $x = 4$ gives $f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{2(2)} = \frac{1}{4}$ .

Derivative of square root using the power rule

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