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Keenan Rhodes

Answered question

2022-04-10

How do you use the limit definition to find the derivative of

f (x) = x^{3} + 1

strasnihwhge

Beginner2022-04-11Added 14 answers

By definition

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

So, with

f (x) = x^{3} + 1

we have:

f^{'} (x) = lim_{h \to 0} \frac{({(x + h)}^{3} + 1) - (x^{3} + 1)}{h}

∴ f^{'} (x) = lim_{h \to 0} \frac{((x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3}) + 1) - x^{3} - 1}{h}

∴ f^{'} (x) = lim_{h \to 0} \frac{x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} + 1 - x^{3} - 1}{h}

∴ f^{'} (x) = lim_{h \to 0} \frac{3 h x^{2} + 3 h^{2} x + h^{3}}{h}

∴ f^{'} (x) = lim_{h \to 0} (3 x^{2} + 3 h x + h^{2})

∴ f^{'} (x) = 3 x^{2}

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