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$Given the function f(x)=-(3sqrtx)/(2), find f^(')(x). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: f^(')(x)=$

Given the function $f(x)=-\frac{3 \sqrt{x}}{2}$ , find $f^{\prime}(x)$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $f^{\prime}(x)=$

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Precalculus

Find trigonometric ratios of special angles

Full solution

Q. Given the function

f(x)=-\frac{3 \sqrt{x}}{2}

, find

f^{\prime}(x)

. Express your answer in radical form without using negative exponents, simplifying all fractions.

\newline

Answer:

f^{\prime}(x)=

Apply Power Rule: To find the derivative of the function $f(x) = -\frac{3\sqrt{x}}{2}$ , we need to apply the power rule for differentiation. The square root of $x$ can be written as $x^{\frac{1}{2}}$ . So, we rewrite the function as $f(x) = -\frac{3}{2}x^{\frac{1}{2}}$ .
Differentiate Using Power Rule: Now, we differentiate the function using the power rule, which states that the derivative of $x^n$ with respect to $x$ is $n\cdot x^{n-1}$ . In this case, $n$ is $\frac{1}{2}$ , so we get $f'(x) = -\left(\frac{3}{2}\right)\cdot\left(\frac{1}{2}\right)\cdot x^{\frac{1}{2} - 1}$ .
Simplify Expression: Simplifying the expression, we get $f'(x) = -\frac{3}{4}x^{-\frac{1}{2}}$ . However, we need to express the answer without using negative exponents.
Express Without Negative Exponents: To express $x^{-1/2}$ without a negative exponent, we write it as $1/\sqrt{x}$ or $1/x^{1/2}$ . Therefore, $f^{\prime}(x) = -\left(\frac{3}{4}\right)\left(\frac{1}{\sqrt{x}}\right)$ .
Combine Constants: Finally, we simplify the fraction by combining the constants. The derivative of the function $f(x) = -\frac{3\sqrt{x}}{2}$ is $f^{\prime}(x) = -\frac{3}{4\sqrt{x}}$ .