▶️ Answer/Explanation
Solution
Correct Answer: A
Step 1: Use the chain rule: \( h(x) = f(g(x)) \Rightarrow h'(x) = f'(g(x)) \cdot g'(x) \)
Step 2: Evaluate \( g(1) \):
\[ g(1) = 2(1) + 1 = 3 \]
Step 3: Compute the slope of \( f(x) \) at \( x = 3 \). The segment from (1, 4) to (4, -2) has a slope of:
\[ \frac{-2 – 4}{4 – 1} = \frac{-6}{3} = -2 \Rightarrow f'(3) = -2 \]
Step 4: Compute \( h'(1) = f'(g(1)) \cdot g'(1) = f'(3) \cdot 2 = -2 \cdot 2 = -4 \)
▶️ Answer/Explanation
Solution
Correct Answer: B
Step 1: Rewrite the function: \[ f(x) = x^{-3} – \frac{1}{x + x^2} \] Step 2: Differentiate each term: \[ \frac{d}{dx}(x^{-3}) = -3x^{-4} \] \[ \frac{d}{dx}\left(\frac{1}{x + x^2}\right) = -\frac{1 + 2x}{(x + x^2)^2} \] Step 3: Combine derivatives: \[ f'(x) = -3x^{-4} + \frac{1 + 2x}{(x + x^2)^2} \] Step 4: Evaluate at \(x = 1\): \[ f'(1) = -3(1)^{-4} + \frac{1 + 2(1)}{(1 + 1^2)^2} = -3 + \frac{3}{4} = -\frac{9}{4} + \frac{3}{4} = -\frac{6}{4} = -1.5 \]
▶️ Answer/Explanation
Solution
Correct Answer: A
Step 1: Differentiate the first term: \[ \frac{d}{dx}\left(\frac{1}{2}x^{4/5}\right) = \frac{1}{2} \cdot \frac{4}{5}x^{-1/5} = \frac{2}{5x^{1/5}} \] Step 2: Differentiate the second term: \[ \frac{d}{dx}\left(-3x^{-5}\right) = 15x^{-6} = \frac{15}{x^6} \] Step 3: Combine the derivatives: \[ \frac{dy}{dx} = \frac{2}{5x^{1/5}} + \frac{15}{x^6} \]
▶️ Answer/Explanation
Solution
Correct Answer: B
Step 1: Rewrite the function in simpler form:
\( f(x) = bx^2 – 13bx + b^2 + x^{-2} \)
Step 2: Differentiate term by term:
\( \frac{d}{dx}(bx^2) = 2bx \)
\( \frac{d}{dx}(-13bx) = -13b \)
\( \frac{d}{dx}(b^2) = 0 \) (since it’s a constant)
\( \frac{d}{dx}(x^{-2}) = -2x^{-3} = -\frac{2}{x^3} \)
Step 3: Combine the results:
\( f'(x) = 2bx – 13b – \frac{2}{x^3} \)