$Limits calculus problem$

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$Limits calculus problem$

Calculus

Notacomputer1235

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Mathe

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I suggest increasing the bounty for this multipart problem.

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Mathe

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I suggest increasing the bounty for this multipart problem.

Answer 1

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Let $f=3x^2$. Then
\[\frac{dy}{dx}(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim_{h \rightarrow 0} \frac{3(x+h)^2-3x^2}{h}\]
\[=\lim_{h \rightarrow 0} \frac{3[(x^2+2xh +h^2)-x^2]}{h}=\lim_{h \rightarrow 0}\frac{3[2x h +h^2]}{h}\]
\[=\lim_{\epsilon \rightarrow 0}3[2x+h]=6x.\]

The answer to the first box is:
\[ \frac{3(x+h)^2-3x^2}{h}\]

Thus \[\frac{dy}{dx}|_{x=7}=6\times 7=42, \text{and} \frac{dy}{dx}|_{x=-5}=6\times (-5)=-30.\]

Erdos

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Erdos

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Let me know if you need any clarifications.
- Notacomputer1235
  
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  what was the answer you put for the first box?
- Erdos
  
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  I updated the solution with the answer to the first box.

$Limits calculus problem$

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