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Question

Find the difference quotient 𝑓(𝑥)−𝑓(3)/𝑥−3 when 𝑓(𝑥)=1+5𝑥−5𝑥^2. Simplify the expression fully as if you were going to compute the limit as 𝑥→3. In particular, cancel common factors of 𝑥−3 in the numerator and denominator if possible. (Use symbolic notation and fractions where needed.)

𝑓(𝑥)−𝑓(3)/𝑥−3 =

Use your result to compute the derivative. (Give your answer as a whole or exact number.)

𝑓′(3)=

Answer 1

f'(x) = 5-10x

f'(3) = 5-10(3) = 5-30= -25

f(x) = 1+5x-5x^2

to take the derivative sum the derivatives of each term

derivative of any constant such as 1 is zero

derivative of 5x is just the coefficient 5,

general rule is take the exponent and multiply it by the coefficient and reduce the exponent by 1

so derivative of 5x is (1)(5)x^1-1 = 5x^0 = 5(1) = 5

and

derivative of -5x^2 = 2(-5)x^2-1 = -10x^1 = -10x

add them up: 0+5-10x = 5-10x

whatever you get taking the limit of the difference quotient has to be the same: 5-10x

then replace x by 3 to get

f'(3) =5-10(3) = 5-30=-25

to get

f'(3) = 5+ 10(3) = 5+30 = 35