24. [-/2 Points] DETAILS MY NOTES SCALCET9 2.8.053. Use the definition of derivative to find \( f^{\prime}(x) \) and \( f^{\prime \prime}(x) \). \( f^{\prime}(x)=\square(x)=4 x^{2}+3 x+2 \) \( f^{\prime \prime}(x)=\square \) Need Help? Read It पatch it

Sure, let's find the first and second derivatives of the function \( f(x) \) using the definition of the derivative. ### Given: \[ f'(x) = 4x^{2} + 3x + 2 \] ### 1. Finding the Second Derivative \( f''(x) \) The second derivative \( f''(x) \) is simply the derivative of the first derivative \( f'(x) \). Let's differentiate \( f'(x) \) with respect to \( x \): \[ f''(x) = \frac{d}{dx} [4x^{2} + 3x + 2] \] Differentiate each term separately: - The derivative of \( 4x^{2} \) is \( 8x \). - The derivative of \( 3x \) is \( 3 \). - The derivative of the constant \( 2 \) is \( 0 \) (since constants do not change). So, combining these results: \[ f''(x) = 8x + 3 \] ### Summary: - **First Derivative:** \[ f'(x) = 4x^{2} + 3x + 2 \] - **Second Derivative:** \[ f''(x) = 8x + 3 \] If you need to derive \( f'(x) \) using the definition of the derivative, here's a brief overview: ### Using the Definition of the Derivative to Find \( f'(x) \) The definition of the derivative is: \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \] Assuming \( f'(x) = 4x^{2} + 3x + 2 \) was obtained using this definition, then the second derivative follows as we've calculated above. If you have the original function \( f(x) \) and need to find \( f'(x) \) from scratch, please provide \( f(x) \), and I can guide you through the process! --- **Final Answers:** \[ f'(x) = 4x^{2} + 3x + 2 \] \[ f''(x) = 8x + 3 \]