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

Question

Evelyn Agard · Accepted Answer

# f'(x) = 3 # The definition of the derivative of #y=f(x)# is # f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h # So Let # f(x) = 3x-7 # then; # \ \ \ \ \ f(x+h) = 3(x+h) -7 # # :. f(x+h) = 3x+3h -7 # And so the derivative of #y=f(x)# is given by: # \ \ \ \ \ f'(x) = lim_(h rarr 0) ( (3x+3h -7) - (3x-7) ) / h # # " " = lim_(h rarr 0) ( 3h ) / h # # " " = lim_(h rarr 0) 3 # # :. f'(x) = 3 #

Ezekiel Alexander · Answer

undefined To find the derivative of $ f(x) = 3x - 7 $ using the limit definition of the derivative, follow these steps:

1. Start with the limit definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = 3x - 7 $ into the formula:
$$ f'(x) = \lim_{h \to 0} \frac{(3(x + h) - 7) - (3x - 7)}{h} $$

3. Expand and simplify the expression inside the limit:
$$ f'(x) = \lim_{h \to 0} \frac{3x + 3h - 7 - 3x + 7}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{3h}{h} $$

4. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \to 0} 3 $$

5. Since the limit of a constant is the constant itself, the derivative is:
$$ f'(x) = 3 $$