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

Answer 1

# f'(x) = 14x #

By definition: # f'(x) = lim_(h->0)(f(x+h)-f(x))/h #
so, with # f(x)=7x^2-3 # we have:
# f'(x) = lim_(h->0)({7(x+h)^2-3}-(7x^2-3))/h #
# = lim_(h->0)({7(x^2+2hx+h^2)-3}-(7x^2-3))/h # # = lim_(h->0)(7x^2+14hx+7h^2-3-7x^2+3)/h # # = lim_(h->0)(color(red)(cancel(7x^2))+14hx+7h^2color(blue)cancel(-3)color(red)cancel(-7x^2)+color(blue)cancel(3))/h # # = lim_(h->0)(14hx+7h^2)/h # # = lim_(h->0)14x+7h # # = 14x #
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Answer 2

To find the derivative of ( f(x) = 7x^2 - 3 ), you can use the definition of a derivative, which states that the derivative of a function ( f(x) ) at a point ( x ) is the limit of the difference quotient as ( h ) approaches 0.

The difference quotient is given by ( \frac{f(x + h) - f(x)}{h} ).

For ( f(x) = 7x^2 - 3 ), the difference quotient becomes ( \frac{7(x + h)^2 - 3 - (7x^2 - 3)}{h} ).

Expanding and simplifying, you get ( \frac{7(x^2 + 2xh + h^2) - 3 - 7x^2 + 3}{h} ).

This simplifies to ( \frac{7x^2 + 14xh + 7h^2 - 7x^2}{h} ).

Further simplification yields ( \frac{14xh + 7h^2}{h} ).

Now, you can cancel out the ( h ) terms in the numerator and denominator to get ( 14x + 7h ).

Now, take the limit of this expression as ( h ) approaches 0 to find the derivative of ( f(x) ).

Taking the limit, you get ( \lim_{h \to 0} (14x + 7h) ).

As ( h ) approaches 0, ( 7h ) approaches 0, so the derivative of ( f(x) ) with respect to ( x ) is ( 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.

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.

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.

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.

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