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Derivative of 1/sqrt(x) using the power rule

Let f(x)=1xf(x) = \dfrac{1}{\sqrt{x}}. Rewrite f(x)f(x) as a power of xx, that is f(x)=x−1/2f(x) = x^{-1/2}, and use the power rule to find f′(x)f'(x).

Hint

Apply the power rule ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1} with n=−12n = -\frac{1}{2}. Be careful subtracting 1 from a negative fraction to get the new exponent.

Step-by-step solution

Rewrite f(x)=1xf(x) = \dfrac{1}{\sqrt{x}} as f(x)=x−1/2f(x) = x^{-1/2}. Applying the power rule ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1} with n=−12n = -\frac{1}{2} gives f′(x)=−12x−1/2−1f'(x) = -\frac{1}{2} x^{-1/2 - 1}.

Simplify the exponent: −12−1=−32-\frac{1}{2} - 1 = -\frac{3}{2}, so f′(x)=−12x−3/2f'(x) = -\frac{1}{2} x^{-3/2}. This can also be written as f′(x)=−12x3/2f'(x) = -\dfrac{1}{2x^{3/2}} or −12x3-\dfrac{1}{2\sqrt{x^3}}.

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