Best solutions to - "How do you find the derivative of f(x)=x3+x2..."

Aedan Berry

Answered question

2022-04-15

How do you find the derivative of

f (x) = x^{3} + x^{2}

using the limit process?

aludirao09au

Beginner2022-04-16Added 11 answers

By definition of the derivative

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

So with

f (x) = x^{3} + x^{2}

we have:

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

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

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

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

∴ f^{'} (x) = 3 x^{2} + 2 x

strasnihwhge

Beginner2022-04-17Added 14 answers

lim_{h \to 0} [\frac{f (x + h) - f (x)}{h}]

lim_{h \to 0} (\frac{{(x + h)}^{3} + {(x + h)}^{2} - (x^{3} + x^{2})}{h})

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

= 3 x^{2} + 2 x

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