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Apuntes

• Derivative of the function: \[f'(x) = \left[\sin(5x) \cdot \sqrt[3]{x^{11}}\right]^3\] Applying the chain rule, the derivative becomes: \[f'(x) = 3 \cdot \left(\sin(5x) \cdot (x+1)^{\frac{3}{2}}\right)^2 \cdot \frac{d}{dx}\left[\sin(5x) \cdot (x+1)^{\frac{3}{2}}\right]\] Next, we differentiate the inner product using the product rule: \[f'(x) = 3 \left(\sin(5x) \cdot (x+1)^{\frac{3}{2}}\right)^2 \left[\sin(5x) \cdot \frac{d}{dx} \left[(x+1)^{\frac{3}{2}}\right] + (x+1)^{\frac{3}{2}} \cdot \frac{d}{dx} \left[\sin(5x)\right]\right]\] Calculating each derivative: \[ \frac{d}{dx} \left[(x+1)^{\frac{3}{2}}\right] = \frac{3}{2} (x+1)^{\frac{1}{2}} \] \[ \frac{d}{dx} \left[\sin(5x)\right] = 5 \cos(5x) \] Substituting back, the derivative simplifies to: \[ f'(x) = 3 \left(\sin(5x) \cdot (x+1)^{\frac{3}{2}}\right)^2 \left[\sin(5x) \cdot \frac{3}{2} (x+1)^{\frac{1}{2}} + (x+1)^{\frac{3}{2}} \cdot 5 \cos(5x)\right] \]