1. Problem: Find the derivative of $F(x) = 5x^3$. Step 1: Recall the power rule for derivatives: $\frac{d}{dx} x^n = nx^{n-1}$. Step 2: Apply the power rule: $$F'(x) = 5 \cdot \frac{d}{dx} x^3 = 5 \cdot 3x^{3-1} = 15x^2$$ 2. Problem: Find the derivative of $F(x) = x^4 + x^3$. Step 1: Use the sum rule and power rule: $$F'(x) = \frac{d}{dx} x^4 + \frac{d}{dx} x^3 = 4x^3 + 3x^2$$ 3. Problem: Find the derivative of $F(x) = \frac{1}{x}$. Step 1: Rewrite $F(x)$ as $x^{-1}$. Step 2: Apply the power rule: $$F'(x) = \frac{d}{dx} x^{-1} = -1 \cdot x^{-2} = -\frac{1}{x^2}$$ 4. Problem: Find the derivative of $F(x) = 4(5x-2)^3$. Step 1: Use the constant multiple rule and chain rule. Step 2: Let $u = 5x - 2$, then $F(x) = 4u^3$. Step 3: Derivative of $u^3$ is $3u^2 \cdot u'$. Step 4: Compute $u' = 5$. Step 5: Apply chain rule: $$F'(x) = 4 \cdot 3u^2 \cdot 5 = 60(5x - 2)^2$$ 5. Problem: Find the derivative of $F(x) = (x^3 + 2) \sqrt{x^4 - x^3}$. Step 1: Rewrite $\sqrt{x^4 - x^3}$ as $(x^4 - x^3)^{1/2}$. Step 2: Use product rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$. Step 3: Let $f(x) = x^3 + 2$, $g(x) = (x^4 - x^3)^{1/2}$. Step 4: Compute $f'(x) = 3x^2$. Step 5: Compute $g'(x)$ using chain rule: $$g'(x) = \frac{1}{2}(x^4 - x^3)^{-1/2} \cdot (4x^3 - 3x^2)$$ Step 6: Apply product rule: $$F'(x) = 3x^2 (x^4 - x^3)^{1/2} + (x^3 + 2) \cdot \frac{1}{2}(x^4 - x^3)^{-1/2} (4x^3 - 3x^2)$$ 6. Problem: Find the derivative of $F(x) = \frac{\sqrt{x^3 + x^2}}{x^3 + x^4}$. Step 1: Rewrite numerator as $(x^3 + x^2)^{1/2}$. Step 2: Use quotient rule: $$F'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$ where $f(x) = (x^3 + x^2)^{1/2}$ and $g(x) = x^3 + x^4$. Step 3: Compute $f'(x)$ using chain rule: $$f'(x) = \frac{1}{2}(x^3 + x^2)^{-1/2} (3x^2 + 2x)$$ Step 4: Compute $g'(x) = 3x^2 + 4x^3$. Step 5: Apply quotient rule: $$F'(x) = \frac{\frac{1}{2}(x^3 + x^2)^{-1/2} (3x^2 + 2x)(x^3 + x^4) - (x^3 + x^2)^{1/2} (3x^2 + 4x^3)}{(x^3 + x^4)^2}$$ 7. Problem: Find the derivative of $F(x) = (\ln x^4)^8$. Step 1: Simplify inside logarithm: $$\ln x^4 = 4 \ln x$$ Step 2: Rewrite $F(x) = (4 \ln x)^8 = 4^8 (\ln x)^8$. Step 3: Use chain rule: $$F'(x) = 4^8 \cdot 8 (\ln x)^7 \cdot \frac{1}{x} = 8 \cdot 4^8 \frac{(\ln x)^7}{x}$$