Questions: Given f(x)=4-2x^2, find f'(x) using the limit definition of the derivative. f'(x)=

Solution Steps

To find the derivative \( f'(x) \) of the function \( f(x) = 4 - 2x^2 \) using the limit definition of the derivative, we use the formula:

\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]

1. Substitute \( f(x) = 4 - 2x^2 \) into the formula.
2. Simplify the expression inside the limit.
3. Take the limit as \( h \) approaches 0.

We start with the function given by

\[ f(x) = 4 - 2x^2. \]

Using the limit definition of the derivative, we have

\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}. \]

Substituting \( f(x) \) into the limit, we calculate

\[ f(x+h) = 4 - 2(x+h)^2 = 4 - 2(x^2 + 2xh + h^2) = 4 - 2x^2 - 4xh - 2h^2. \]

Thus,

\[ f(x+h) - f(x) = (4 - 2x^2 - 4xh - 2h^2) - (4 - 2x^2) = -4xh - 2h^2. \]

Now, substituting this back into the limit gives us

\[ f'(x) = \lim_{{h \to 0}} \frac{-4xh - 2h^2}{h} = \lim_{{h \to 0}} (-4x - 2h). \]

Taking the limit as \( h \) approaches 0, we find

\[ f'(x) = -4x. \]

Final Answer