Hello there,
This problem originally asks for it to be solved using the difference quotient, but since it is asking for a derivative of a specific number, I am trying to do it with the alternative difference quotient formula.
Thank you.
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1. Evaluate the derivative \(\displaystyle f'(-3)\), if the function is \(\displaystyle f(x) = x^3 - 2x\).
\(\displaystyle f'(c) = \lim_{x \to c}\frac {f(x) - f(c)}{x - c}\)
\(\displaystyle f'(-3) = \lim_{x \to -3}\frac {f(x) - f(-3)}{x + 3}\)
\(\displaystyle f'(-3) = \lim_{x \to c}\frac {x^3 - 2x - (-21)}{x + 3}\)
\(\displaystyle f'(-3) = \lim_{x \to c}\frac {(x - 3)(x^2 - 3x + 7)}{x + 3}\)
However, I do not see how I can simplify this fraction even more. The quadratic equation in the numerator cannot be factored as its roots are irrational.
This problem originally asks for it to be solved using the difference quotient, but since it is asking for a derivative of a specific number, I am trying to do it with the alternative difference quotient formula.
Thank you.
---
1. Evaluate the derivative \(\displaystyle f'(-3)\), if the function is \(\displaystyle f(x) = x^3 - 2x\).
\(\displaystyle f'(c) = \lim_{x \to c}\frac {f(x) - f(c)}{x - c}\)
\(\displaystyle f'(-3) = \lim_{x \to -3}\frac {f(x) - f(-3)}{x + 3}\)
\(\displaystyle f'(-3) = \lim_{x \to c}\frac {x^3 - 2x - (-21)}{x + 3}\)
\(\displaystyle f'(-3) = \lim_{x \to c}\frac {(x - 3)(x^2 - 3x + 7)}{x + 3}\)
However, I do not see how I can simplify this fraction even more. The quadratic equation in the numerator cannot be factored as its roots are irrational.