How do you use the definition of the derivative to differentiate the function #f(x)= 2x^2-5#?

To differentiate the function ( f(x) = 2x^2 - 5 ) using the definition of the derivative, follow these steps:

1. Start with the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

2. Substitute the function ( f(x) = 2x^2 - 5 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(2(x+h)^2 - 5) - (2x^2 - 5)}{h} ]

3. Expand and simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{2(x^2 + 2hx + h^2) - 5 - 2x^2 + 5}{h} ] [ f'(x) = \lim_{h \to 0} \frac{2x^2 + 4hx + 2h^2 - 2x^2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{4hx + 2h^2}{h} ]

4. Factor out ( h ) from the numerator: [ f'(x) = \lim_{h \to 0} \frac{h(4x + 2h)}{h} ]

5. Cancel out the ( h ) terms: [ f'(x) = \lim_{h \to 0} (4x + 2h) ]

6. Evaluate the limit as ( h ) approaches 0: [ f'(x) = 4x ]

Therefore, the derivative of the function ( f(x) = 2x^2 - 5 ) with respect to ( x ) is ( f'(x) = 4x ).