calculate the derivative by the limit definition: f(x) = 6x^3 + 2

To calculate the derivative of the function \(f(x) = 6x^3 + 2\) using the limit definition of a derivative, you can use the following formula: \[f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}\] In this case, \(f(x) = 6x^3 + 2\), and we want to find f'(x). Now, plug this into the limit definition formula: \[f'(x) = \lim_{{h \to 0}} \frac{(6(x + h)^3 + 2) - (6x^3 + 2)}{h}\] Let's simplify the expression inside the limit: \[f'(x) = \lim_{{h \to 0}} \frac{6(x^3 + 3x^2h + 3xh^2 + h^3) + 2 - 6x^3 - 2}{h}\] Now, we can cancel out the terms that will simplify: \[f'(x) = \lim_{{h \to 0}} \frac{6(3x^2h + 3xh^2 + h^3)}{h}\] Next, factor out an \(h\) from the numerator: \[f'(x) = \lim_{{h \to 0}} \frac{h(18x^2 + 18xh + 6h^2)}{h}\] Now, cancel out the common factor of \(h\) in the numerator and denominator: \[f'(x) = \lim_{{h \to 0}} 18x^2 + 18xh + 6h^2\] Now, we can calculate the limit as \(h\) approaches 0: \[f'(x) = 18x^2 + 18x(0) + 6(0)^2\] \[f'(x) = 18x^2\] So, the derivative of \(f(x) = 6x^3 + 2\) with respect to x is \(f'(x) = 18x^2\).