Resolved: "Using the limit definition, how do you differentiate..."

Noe Wade

Answered question

2022-04-21

Using the limit definition, how do you differentiate

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

Alexis Wolf

Beginner2022-04-22Added 13 answers

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

We know that

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

and

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

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

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

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

= lim_{h \to 0} \frac{- 2 h x - h^{2}}{h}

= lim_{h \to 0} - 2 x - h

Now we can plug in 0 for h.
=-2x
Thus, f'(x)=-2.

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