How do you find the derivative of #-5x^2+8x+2# using the limit definition?

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Answer 1 · 2016-09-05T18:47:29

#8 - 10 x#

We have: #f(x) = - 5 x^(2) + 8 x + 2#

Let's evaluate the derivative of this expression by differentiating using first principles:

#=> f'(x) = lim_(h -> 0) ((f(x + h) - f(x)) / (h))#

#=> f'(x) = lim_(h -> 0) (((- 5 (x + h)^(2) + 8 (x + h) + 2) - (- 5 x^(2) + 8 x + 2)) / (h))#

#=> f'(x) = lim_(h -> 0) ((- 5 (x^(2) + 2 h x + h^(2)) + 8 x + 8 h + 2 + 5 x^(2) - 8 x - 2) / (h))#

#=> f'(x) = lim_(h -> 0) ((- 5 x^(2) - 10 h x - 5 h^(2) + 8 h + 5 x^(2)) / (h))#

#=> f'(x) = lim_(h -> 0) ((8 h - 10 h x - 5 h^(2)) / (h))#

#=> f'(x) = lim_(h -> 0) (8 - 10 x - 5 h)#

#=>f'(x) = 8 - 10 x#