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Given the function f(x)=-(4)/(sqrtx), find f^(')(x). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: f^(')(x)=

Given the function f(x)=−4x f(x)=-\frac{4}{\sqrt{x}} , find f′(x) f^{\prime}(x) . Express your answer in radical form without using negative exponents, simplifying all fractions.\newlineAnswer: f′(x)= f^{\prime}(x)=

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Q. Given the function f(x)=−4x f(x)=-\frac{4}{\sqrt{x}} , find f′(x) f^{\prime}(x) . Express your answer in radical form without using negative exponents, simplifying all fractions.\newlineAnswer: f′(x)= f^{\prime}(x)=
  1. Rewrite Function: To find the derivative of the function f(x)=−4xf(x) = -\frac{4}{\sqrt{x}}, we need to apply the chain rule and the power rule for differentiation. The function can be rewritten as f(x)=−4x−12f(x) = -4x^{-\frac{1}{2}}.
  2. Apply Power Rule: Differentiating −4x(−1/2)-4x^{(-1/2)} with respect to xx, we use the power rule which states that ddx[xn]=n⋅x(n−1)\frac{d}{dx} [x^n] = n \cdot x^{(n-1)}. Here, n=−1/2n = -1/2, so we get f′(x)=−4⋅(−1/2)⋅x(−1/2−1)f^{\prime}(x) = -4 \cdot (-1/2) \cdot x^{(-1/2 - 1)}.
  3. Simplify Expression: Simplifying the expression, we have f′(x)=2×x(−32)f^{'}(x) = 2 \times x^{(-\frac{3}{2})}.
  4. Convert to Radical Form: To express the answer in radical form without using negative exponents, we rewrite x−32x^{-\frac{3}{2}} as 1x32\frac{1}{x^{\frac{3}{2}}} which is the same as 1x3\frac{1}{\sqrt{x}^3}.
  5. Convert to Radical Form: To express the answer in radical form without using negative exponents, we rewrite x−32x^{-\frac{3}{2}} as 1x32\frac{1}{x^{\frac{3}{2}}} which is the same as 1x3\frac{1}{\sqrt{x}^3}.Therefore, the derivative of the function in radical form is f′(x)=2x3f^{\prime}(x) = \frac{2}{\sqrt{x}^3}.

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