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Find the derivative of the function f(x) = 7x - 3 by the limit process.

User Ectype
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Final answer:

The derivative of the function f(x) = 7x - 3, found by the limit process, is f'(x) = 7. This is determined by applying the definition of the derivative, simplifying the algebraic expression, and recognizing that the derivative of a constant is zero.

Step-by-step explanation:

To find the derivative of the function f(x) = 7x - 3 by the limit process, we employ the definition of the derivative, which is:

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

Applying this definition to our function, we get:

f'(x) = lim (h → 0) [((7(x+h) - 3) - (7x - 3))/h]

Simplifying the expression inside the limit, we have:

f'(x) = lim (h → 0) [(7x + 7h - 3 - 7x + 3)/h]

The terms 7x and -7x cancel out, and the constants -3 and +3 cancel each other as well:

f'(x) = lim (h → 0) [7h/h]

Now, we can simplify by canceling h:

f'(x) = lim (h → 0) [7]

Since 7 is a constant and does not depend on h, we can conclude:

f'(x) = 7

User Heltonbiker
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