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What is the value of (d)/(dx)(sqrtx) at x=9 ?

What is the value of ddx(x) \frac{d}{d x}(\sqrt{x}) at x=9 x=9 ?

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Full solution

Q. What is the value of ddx(x) \frac{d}{d x}(\sqrt{x}) at x=9 x=9 ?
  1. Identify Function: Identify the function whose derivative we need to find.\newlineThe function is f(x)=xf(x) = \sqrt{x}, which can also be written as f(x)=x(1/2)f(x) = x^{(1/2)}.
  2. Apply Power Rule: Apply the power rule for differentiation to find the derivative of f(x)=x12f(x) = x^{\frac{1}{2}}. The power rule states that if f(x)=xnf(x) = x^n, then f′(x)=n⋅xn−1f'(x) = n \cdot x^{n-1}. Using this rule, the derivative of f(x)=x12f(x) = x^{\frac{1}{2}} is f′(x)=12⋅x(12−1)=12⋅x−12f'(x) = \frac{1}{2} \cdot x^{\left(\frac{1}{2}-1\right)} = \frac{1}{2} \cdot x^{-\frac{1}{2}}.
  3. Simplify Derivative: Simplify the expression for the derivative. f′(x)=12x−12f'(x) = \frac{1}{2}x^{-\frac{1}{2}} can be rewritten as f′(x)=12(1x)f'(x) = \frac{1}{2}\left(\frac{1}{\sqrt{x}}\right).
  4. Evaluate at x=9x=9: Evaluate the derivative at x=9x=9.\newlineSubstitute xx with 99 in the expression f′(x)=12⋅1xf'(x) = \frac{1}{2}\cdot\frac{1}{\sqrt{x}}.\newlinef′(9)=12⋅19=12⋅13=16f'(9) = \frac{1}{2}\cdot\frac{1}{\sqrt{9}} = \frac{1}{2}\cdot\frac{1}{3} = \frac{1}{6}.

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