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How do you use the limit definition of the derivative to find the derivative of #f(x)=5x+1#?

Answer 1

#f'(x) = 5#

Limit definition of derivative: #f'(x) = limh->0 ((f(x+h)-f(x))/h)#

Plugging in our function #f(x) = 5x+1# #f'(x) = limh->0 [((5(x+h)+1)-(5x+1))/h]#

Distributing terms in the numerator: #f'(x) = limh->0 [(5x+5h+1-5x-1)/h]#

Cancel out terms in the numerator: #f'(x) = limh->0 [(cancel(5x)+5hcancel(+1)cancel(-5x)cancel(-1))/h]# #f'(x)=limh->0[(5cancel(h))/cancel(h)]# #f'(x)=5#

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Answer 2

Isaiah Allen

To use the limit definition of the derivative to find the derivative of ( f(x) = 5x + 1 ), follow these steps:

Start with the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
Substitute the given function ( f(x) = 5x + 1 ) into the formula.
Expand ( f(x + h) ) using the given function.
Subtract ( f(x) ) from ( f(x + h) ).
Simplify the expression.
Take the limit as ( h ) approaches 0 to find the derivative.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.