Find the slope of the function's graph at the given point. T \( f(x)=\sqrt{3 x+7},(3,4) \) The slope of the function's graph at \( (3,4) \) is (Simplify your answer.)

**Step 1: Write the function in exponent form** Given the function \[ f(x)=\sqrt{3x+7}=(3x+7)^{\frac{1}{2}}, \] we can differentiate it using the chain rule. --- **Step 2: Differentiate using the chain rule** Differentiating with respect to \( x \), we have \[ f'(x)=\frac{1}{2}(3x+7)^{-\frac{1}{2}} \cdot \frac{d}{dx}(3x+7). \] The derivative of \( 3x+7 \) is \( 3 \). Thus, \[ f'(x)=\frac{1}{2}\cdot 3\cdot (3x+7)^{-\frac{1}{2}}=\frac{3}{2}(3x+7)^{-\frac{1}{2}}. \] Writing the negative exponent as a denominator gives \[ f'(x)=\frac{3}{2\sqrt{3x+7}}. \] --- **Step 3: Evaluate the derivative at \( x=3 \)** Substitute \( x=3 \) into the derivative: \[ f'(3)=\frac{3}{2\sqrt{3(3)+7}}. \] Calculate inside the square root: \[ 3(3)+7=9+7=16. \] So, \[ f'(3)=\frac{3}{2\sqrt{16}}=\frac{3}{2\cdot 4}=\frac{3}{8}. \] --- **Step 4: State the final answer** The slope of the function's graph at the point \( (3,4) \) is \[ \frac{3}{8}. \]