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Haylee Bowen

Answered question

2022-04-11

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

f (x) = - 6 - x^{2}

at x=7?

tutaonana223a

Beginner2022-04-12Added 15 answers

For a function f(x), its derivative

\frac{d f}{d x}

is defined as

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

f (x) = - 6 - x^{2}, f (x + h) = - 6 - {(x + h)}^{2}

and

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

= lim_{h \to 0} \frac{- 6 - {(x + h)}^{2} - (- 6 - x^{2})}{h}

= lim_{h \to 0} \frac{- 6 - x^{2} - 2 h x - h^{2} + 6 + x^{2}}{h}

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

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

=-2x
and at x=7, we have

\frac{d f}{d x} = - 2 \times 7 = - 14

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