12. [0/1 Points] DETAILS MYNOTES LARCALC 122.4 .085 P Find the second derivative of the function: \[ \begin{array}{c} f(x)=\sin \left(8 x^{2}\right) \\ f^{\prime}(x)=4 \cos \left(2 x^{2}\right)=16 x^{2} \sin \left(2 x^{2}\right) \end{array} \] Need Holp? \( \square \) Read It

Find the second derivative of the function \( f(x)=\sin(8x^2) \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(f^{\prime}\left(x\right)=\frac{d}{dx}\left(\sin\left(8x^{2}\right)\right)\) - step1: Use differentiation rules: \(f^{\prime}\left(x\right)=\frac{d}{dg}\left(\sin\left(g\right)\right)\times \frac{d}{dx}\left(8x^{2}\right)\) - step2: Find the derivative: \(f^{\prime}\left(x\right)=\cos\left(g\right)\times \frac{d}{dx}\left(8x^{2}\right)\) - step3: Calculate: \(f^{\prime}\left(x\right)=\cos\left(g\right)\times 16x\) - step4: Substitute back: \(f^{\prime}\left(x\right)=\cos\left(8x^{2}\right)\times 16x\) - step5: Multiply the terms: \(f^{\prime}\left(x\right)=16x\cos\left(8x^{2}\right)\) - step6: Evaluate: \(f^{\prime\prime}\left(x\right)=\frac{d}{dx}\left(16x\cos\left(8x^{2}\right)\right)\) - step7: Use differentiation rules: \(f^{\prime\prime}\left(x\right)=\frac{d}{dx}\left(16\right)\times x\cos\left(8x^{2}\right)+16\times \frac{d}{dx}\left(x\right)\times \cos\left(8x^{2}\right)+16x\times \frac{d}{dx}\left(\cos\left(8x^{2}\right)\right)\) - step8: Find the derivative: \(f^{\prime\prime}\left(x\right)=0\times x\cos\left(8x^{2}\right)+16\times \frac{d}{dx}\left(x\right)\times \cos\left(8x^{2}\right)+16x\times \frac{d}{dx}\left(\cos\left(8x^{2}\right)\right)\) - step9: Calculate: \(f^{\prime\prime}\left(x\right)=0+16\times \frac{d}{dx}\left(x\right)\times \cos\left(8x^{2}\right)+16x\times \frac{d}{dx}\left(\cos\left(8x^{2}\right)\right)\) - step10: Calculate: \(f^{\prime\prime}\left(x\right)=0+16\times 1\times \cos\left(8x^{2}\right)+16x\times \frac{d}{dx}\left(\cos\left(8x^{2}\right)\right)\) - step11: Calculate: \(f^{\prime\prime}\left(x\right)=0+16\cos\left(8x^{2}\right)+16x\times \frac{d}{dx}\left(\cos\left(8x^{2}\right)\right)\) - step12: Calculate: \(f^{\prime\prime}\left(x\right)=0+16\cos\left(8x^{2}\right)+16x\left(-16x\sin\left(8x^{2}\right)\right)\) - step13: Calculate: \(f^{\prime\prime}\left(x\right)=0+16\cos\left(8x^{2}\right)-16^{2}x^{2}\sin\left(8x^{2}\right)\) - step14: Calculate: \(f^{\prime\prime}\left(x\right)=16\cos\left(8x^{2}\right)-256x^{2}\sin\left(8x^{2}\right)\) The second derivative of the function \( f(x) = \sin(8x^2) \) is \( f''(x) = 16\cos(8x^2) - 256x^2\sin(8x^2) \).