Question

find the derivative of $x^{\frac{1}{2}} - x^{-\frac{1}{2}}$

Explanation:

Step1: Recall power - rule

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$.

Step2: Differentiate each term

For the first term $y_1 = x^{\frac{1}{2}}$, using the power - rule, $\frac{d}{dx}(x^{\frac{1}{2}})=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}$.
For the second term $y_2=-x^{-\frac{1}{2}}$, using the power - rule, $\frac{d}{dx}(-x^{-\frac{1}{2}})=-(-\frac{1}{2})x^{-\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{3}{2}}$.

Step3: Combine the derivatives

The derivative of $y = x^{\frac{1}{2}}-x^{-\frac{1}{2}}$ is $\frac{dy}{dx}=\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{3}{2}}$.

Answer:

$\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{3}{2}}$