▶️ Answer/Explanation
Solution
Correct Answer: A
Step 1: Find the derivative \( f'(x) \): \[ f(x) = \frac{4}{x} + 5x – 1 = 4x^{-1} + 5x – 1 \] \[ f'(x) = -4x^{-2} + 5 = -\frac{4}{x^2} + 5 \] Step 2: Evaluate at \( x = 2 \): \[ f'(2) = -\frac{4}{(2)^2} + 5 = -\frac{4}{4} + 5 = -1 + 5 = 4 \]
▶️ Answer/Explanation
Solution
Correct Answer: C
Step 1: Rewrite the function: \[ f(x) = \sqrt{x} = x^{1/2} \] Step 2: Find the derivative: \[ f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \] Step 3: Evaluate at \( x = 5 \): \[ f'(5) = \frac{1}{2\sqrt{5}} \]
▶️ Answer/Explanation
Solution
Correct Answer: C
Step 1: Find the derivative using the power rule: \[ f(x) = x^{\frac{3}{2}} \Rightarrow f'(x) = \frac{3}{2}x^{\frac{1}{2}} = \frac{3}{2}\sqrt{x} \] Step 2: Evaluate at \( x = 4 \): \[ f'(4) = \frac{3}{2}\sqrt{4} = \frac{3}{2} \times 2 = 3 \]
▶️ Answer/Explanation
Solution
Correct Answer: A
Step 1: Rewrite the function using exponents: \[ f(x) = x^{1/2} + 3x^{-1/2} \] Step 2: Find the derivative using the power rule: \[ f'(x) = \frac{1}{2}x^{-1/2} – \frac{3}{2}x^{-3/2} \] \[ f'(x) = \frac{1}{2\sqrt{x}} – \frac{3}{2x^{3/2}} \] Step 3: Evaluate at \( x = 4 \): \[ f'(4) = \frac{1}{2\sqrt{4}} – \frac{3}{2(4)^{3/2}} = \frac{1}{4} – \frac{3}{16} = \frac{4}{16} – \frac{3}{16} = \frac{1}{16} \]