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Using the function f(x) = -3/X

a.) Find the derivative of the function at x = 2. Use the definition of
derivative.
b.) Find the equation of the tangent line at x=2

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a) To find the derivative of the function f(x) = -3/x at x = 2 using the definition of derivative, we can start by applying the limit definition of the derivative:

f'(x) = lim(h->0) [(f(x + h) - f(x))/h]

Substituting the given function:

f'(x) = lim(h->0) [(-3/(x + h) - (-3/x))/h]

Simplifying the expression:

f'(x) = lim(h->0) [-3(x - (x + h))/(x(x + h)h)]

f'(x) = lim(h->0) [-3(-h)/(x(x + h)h)]

f'(x) = lim(h->0) [3/(x(x + h))]

Now, substitute x = 2 into the expression:

f'(2) = lim(h->0) [3/(2(2 + h))]

Simplifying further:

f'(2) = lim(h->0) [3/(2(2 + h))]
= 3/(2(2)) (since h -> 0, we can substitute h with 0 in the denominator)
= 3/4

Therefore, the derivative of the function f(x) = -3/x at x = 2 is f'(2) = 3/4.

b) To find the equation of the tangent line at x = 2, we can use the point-slope form of a linear equation. We already have the slope, which is the derivative f'(2) = 3/4, and we need a point on the line. We can use the point (2, f(2)).

Substituting x = 2 into the original function:

f(2) = -3/2 = -1.5

So, the point on the line is (2, -1.5).

Using the point-slope form of a linear equation:

y - y1 = m(x - x1)

Substituting the values:

y - (-1.5) = (3/4)(x - 2)

Simplifying:

y + 1.5 = (3/4)(x - 2)

y = (3/4)x - (3/2) - (3/2)

y = (3/4)x - 3/2

Therefore, the equation of the tangent line at x = 2 is y = (3/4)x - 3/2.
User Danila Alpatov
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