Questions: Let f(x) = 3/sqrt(x) + 4. Then f'(x) is 3/(2 x^(3/2)) -3/(2 x^(3/2)) + 4 -3/(2 x^(3/2)) 3^(2 x)/(x^(3/2)) (x^(3/2))/(2 x^(3/2)) + 4

Transcript text: Let $f(x)=\frac{3}{\sqrt{x}}+4$. Then $f^{\prime}(x)$ is $\frac{3}{2 x^{\frac{3}{2}}}$ $-\frac{3}{2 x^{\frac{3}{2}}}+4$ $-\frac{3}{2 x^{\frac{3}{2}}}$ $\frac{3^{2 x}}{x^{\frac{3}{2}}}$ $\frac{x^{\frac{3}{2}}}{2 x^{\frac{3}{2}}}+4$

Solution

Solution Steps

Step 1: Define the Function

We start with the function defined as \[ f(x) = \frac{3}{\sqrt{x}} + 4. \]

Step 2: Rewrite the Function

To facilitate differentiation, we rewrite the function using exponents: \[ f(x) = 3x^{-\frac{1}{2}} + 4. \]

Step 3: Differentiate the Function

Next, we apply the power rule to differentiate $ f(x) $: \[ f^{\prime}(x) = \frac{d}{dx}(3x^{-\frac{1}{2}}) + \frac{d}{dx}(4). \] The derivative of the constant $ 4 $ is $ 0 $, and using the power rule, we find: \[ \frac{d}{dx}(3x^{-\frac{1}{2}}) = -\frac{3}{2}x^{-\frac{3}{2}}. \]

Step 4: Combine the Results

Thus, the derivative of the function is: \[ f^{\prime}(x) = -\frac{3}{2}x^{-\frac{3}{2}} + 0 = -\frac{3}{2\sqrt{x^3}}. \]

Step 5: Final Expression

We can express the derivative in a simplified form: \[ f^{\prime}(x) = -\frac{3}{2x^{\frac{3}{2}}}. \]