Finding the Derivative of gx = 7x Using the Definition
The derivative of a function provides insight into its rate of change at any point. To find the derivative of the function g(x) = 7x using the definition of the derivative, we can use the limit definition:
f'(x) = lim (h -> 0) [(f(x + h) – f(x)) / h]
Let’s apply this definition step-by-step:
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Evaluate g(x + h):
g(x + h) = 7(x + h) = 7x + 7h
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Evaluate g(x):
g(x) = 7x
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Substitute into the definition of the derivative:
f'(x) = lim (h -> 0) [(g(x + h) – g(x)) / h]
= lim (h -> 0) [(7x + 7h – 7x) / h]
= lim (h -> 0) [7h / h] -
Simplify the expression:
= lim (h -> 0) [7]
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Take the limit:
= 7
Thus, the derivative of the function g(x) = 7x is:
g'(x) = 7
This means that the function g(x) has a constant rate of change of 7 for all values of x. This is consistent with the fact that linear functions have constant derivatives.