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tatiyna
2 years ago
15

Use the limit definition of the derivative to find the instantaneous rate of change of

Mathematics
1 answer:
Lynna [10]2 years ago
8 0

f'(4) = 43

Explanation:

Given: f(x)=5x^2 + 3x + 3

\displaystyle f'(x)= \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}

Note that

f(x+h) = 5(x+h)^2 + 3(x+h) + 3

\:\:\;\:\:\:\:= 5(x^2 + 2hx + h^2) + 3x + 3h +3

\:\:\;\:\:\:\:= 5x^2 + 10hx + 5h^2 + 3x + 3h +3

Substituting the above equation into the expression for f'(x), we can then write f'(x) as

\displaystyle f'(x) = \lim_{h \to 0} \dfrac{10hx + 3h + 5h^2}{h}

\displaystyle\:\:\;\:\:\:\:= \lim_{h \to 0} (10x +3 +5h)

\:\:\;\:\:\:\:= 10x + 3

Therefore,

f'(4) = 10(4) + 3 = 43

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