Use limits to find f' (x) if f(x) = 7x.

Guida verificata passo dopo passo

Step 1: Recall the definition of the derivative using limits. The derivative of a function f(x) at a point x is given by the limit: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).

Step 2: Substitute the given function f(x) = 7x into the derivative definition. This gives us: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{7(x+h) - 7x}{h}\).

Step 3: Simplify the expression inside the limit. Distribute the 7 in the numerator: 7(x+h) = 7x + 7h. So, the expression becomes: \(\frac{7x + 7h - 7x}{h}\).

Step 4: Cancel out the terms in the numerator. The 7x terms cancel each other, leaving: \(\frac{7h}{h}\).

Step 5: Simplify the fraction by canceling h in the numerator and denominator, resulting in 7. Therefore, f'(x) = \(\lim\)_{h \(\to\) 0} 7 = 7.

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