Given \( f(x)=4-2 x^{2} \), find \( f^{\prime}(x) \) using the limit definition of the derivative. \( f^{\prime}(x)=\square \)

To find the derivative \( f'(x) \) using the limit definition, we start with the formula: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Substituting \( f(x) = 4 - 2x^2 \): 1. Calculate \( f(x+h) \): \[ f(x+h) = 4 - 2(x+h)^2 = 4 - 2(x^2 + 2xh + h^2) = 4 - 2x^2 - 4xh - 2h^2 \] 2. Now, substitute into the limit: \[ f'(x) = \lim_{h \to 0} \frac{(4 - 2x^2 - 4xh - 2h^2) - (4 - 2x^2)}{h} \] \[ = \lim_{h \to 0} \frac{-4xh - 2h^2}{h} \] \[ = \lim_{h \to 0} (-4x - 2h) \] As \( h \to 0 \): \[ f'(x) = -4x \] Thus, \( f'(x) = -4x \). --- The concept of derivatives was first formulated by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, laying the groundwork for calculus. Their work helped explain motion, change, and rates, revolutionizing mathematics and physics. Today's applications range from predicting the motion of planets to optimizing business processes, illustrating the crossover from theory to practical impact. In the context of limits, it's essential to remember that the expression should always be simplified before taking the limit. A common mistake is to overlook canceling terms or mishandling the limit as \( h \) approaches zero. Always double-check that you are not dividing by zero and that your final expression is in its simplest form!