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

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Answer to a math question calculate the derivative by the limit definition: f$x$ = 6x^3 + 2

Frederik
4.6
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$.
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