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How do you find the derivative of #f(x) = 7x^2 - 3 # using the limit definition?

Answer 1

Elijah Abram

#14x#

f#'(x) = lim_{h \to 0} ( (7(x+h)^2 - 3) - (7x^2 - 3))/h#

# = lim_{h \to 0} ( (7(x^2+2xh + h^2) - 3) - (7x^2 - 3))/h#

# = lim_{h \to 0} ( 7x^2+14xh + 7h^2 - 3 - 7x^2 + 3)/h#

# = lim_{h \to 0} ( 14xh + 7h^2 )/h#

# = lim_{h \to 0} 14x + 7h = 14x#

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Answer 2

Cameron Alexander

To find the derivative of ( f(x) = 7x^2 - 3 ) using the limit definition, we use the formula:

[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} ]

First, we need to calculate ( f(x + h) ):

[ f(x + h) = 7(x + h)^2 - 3 ] [ f(x + h) = 7(x^2 + 2xh + h^2) - 3 ] [ f(x + h) = 7x^2 + 14xh + 7h^2 - 3 ]

Now, we can substitute into the formula:

[ f'(x) = \lim_{{h \to 0}} \frac{{7x^2 + 14xh + 7h^2 - 3 - (7x^2 - 3)}}{h} ] [ f'(x) = \lim_{{h \to 0}} \frac{{7x^2 + 14xh + 7h^2 - 3 - 7x^2 + 3}}{h} ] [ f'(x) = \lim_{{h \to 0}} \frac{{14xh + 7h^2}}{h} ] [ f'(x) = \lim_{{h \to 0}} (14x + 7h) ] [ f'(x) = 14x ]

So, the derivative of ( f(x) = 7x^2 - 3 ) is ( f'(x) = 14x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.