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Vikki [24]
2 years ago
6

If f(x) = 2x^2 + 4x + 7, find f'(-5), using the definition of derivative. f'(-5) is the limit as x → -5

Mathematics
1 answer:
rosijanka [135]2 years ago
3 0

Using the limit definition of the derivative, you have

\displaystyle f'(-5) = \lim_{x\to-5} \frac{f(x) - f(-5)}{x - (-5)} = \lim_{x\to-5} \frac{(2x^2+4x+7) - 37}{x + 5}

Simplify the numerator:

(2<em>x</em> ² + 4<em>x</em> + 7) - 37 = 2<em>x</em> ² + 4<em>x</em> - 30

… = 2 (<em>x</em> ² + 2<em>x</em> - 15)

… = 2 (<em>x</em> + 5) (<em>x</em> - 3)

Then

\displaystyle f'(-5) = \lim_{x\to-5}\frac{2(x+5)(x-3)}{x+5} = \lim_{x\to-5} 2(x-3) = \boxed{-16}

• • •

For your second question in the comments, if <em>f(x)</em> = -2<em>x</em> ² + 3<em>x</em> - 7, then by the definition of the derivative, you have

\displaystyle f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}h = \lim_{h\to0}\frac{(-2(x+h)^2+3(x+h)-7) - (-2x^2 + 3x - 7)}h

Simplify the numerator:

(-2 (<em>x</em> + <em>h</em>)² + 3 (<em>x</em> + <em>h</em>) - 7) - (-2<em>x</em> ² + 3<em>x</em> - 7)

… = (-2<em>x</em> ² - 4<em>xh</em> - 2<em>h</em> ² + 3<em>x</em> + 3<em>h</em> - 7) - (-2<em>x</em> ² + 3<em>x</em> - 7)

… = -4<em>xh</em> - 2<em>h</em> ² + 3<em>h</em>

Now compute the limit:

\displaystyle f'(x) = \lim_{h\to0}\frac{-4xh-2h^2+3h}h = \lim_{h\to0}(-4x-2h+3) = \boxed{-4x+3}

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