Question

What is the derivative of 3x^2 + 5?

Answer 1

6x

Answer 2

Chris did you want the formal definition of derivative to be used?
\[f'(x)=\lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f( x)}{\Delta x}=\lim_{\Delta x \rightarrow 0}\frac{[3(x+\Delta x)^2+5]-[3x^2+5]}{\Delta x}\]
\[=\lim_{\Delta x \rightarrow 0}\frac{3(x^2+ 2 x \Delta x+(\Delta x)^2)+5-3x^2-5}{\Delta x}\]
\[=\lim_{\Delta x \rightarrow 0}\frac{3x^2-3x^2+5-5+6 x \Delta x+3 (\Delta x)^2}{\Delta x}\]
\[=\lim_{\Delta x \rightarrow 0}\frac{6x \Delta x+3(\Delta x)^2}{\Delta x}\]
\[=\lim_{\Delta x \rightarrow 0}\Delta x \cdot \frac{6x +3 \Delta x}{\Delta x}\]
\[=\lim_{\Delta x \rightarrow 0}(6x+3 \Delta x)=6x+3(0)+6x\]

Answer 3

:) I like that definition. It is so cute.

Answer 4

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