functionsUse the first principle to determine f'(3) whether f(x)=-5x^2.

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Calculus

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justaguide's profile pic

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First find the derivative of f(x)=-5x^2. From first principles this is equal to lim h-->0 [(f(x + h) - f(x))/h]

f(x) = -5x^2

lim h-->0 [(-5(x + h)^2 + 5x^2)/h]

=> -5*lim h-->0 [((x + h)^2 - x^2)/h]

=> -5*lim h-->0 [((x + h)^2 - x^2)/(x + h - x)]

=> -5*lim h-->0 [((x + h + x)(x + h - x))/(x + h - x)]

=> -5*lim h-->0 [(x + h + x)]

substitute h = 0

=> -5*2x

=> -10x

f'(3 ) = -10*3

=> -30

The required value of f'(3) = -30

giorgiana1976's profile pic

Posted on

lim [f(x) - f(3)]/(x-3), for x->3

lim (-5x^2 + 45)/(x - 3) = lim -5(x^2 - 9)/(x-3)

Since x^2 - 9 = (x-3)(x+3, we'll have:

lim -5(x^2 - 9)/(x-3) = -5lim (x-3)(x+3)/(x-3)

We'll simplify by (x-3) and we'll get:

-5lim (x-3)(x+3)/(x-3) = -5lim(x+3)

We'll substitute x by 3 in the expression of limit and we'll get:

-5lim(x+3) = -5(3+3) = -30

Therefore, f'(3) = -30, using the first principle for calculating the derivative.

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