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How do you use the limit definition of the derivative to find the derivative of #f(x)=7x+1#?

Answer 1

Evelyn Agard

#f'(x) = 7#

Ok we know from the power rule that we expect #f'(x) = 7# this allows us to check that we've got the answer correct.

#f'(x) = lim_(hrarr0) (f(x+h) - f(x))/h#

#=lim_(hrarr0)(7(x+h) + 1 - (7x+1))/h#

#=lim_(hrarr0) (7x+7h+1 - 7x - 1)/h = lim_(hrarr0) (7h)/h = lim_(hrarr0)7 = 7#

This is what we expected.

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Answer 2

William Abdalla

To use the limit definition of the derivative to find the derivative of ( f(x) = 7x + 1 ), follow these steps:

Write down the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
Substitute the function ( f(x) = 7x + 1 ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{(7(x + h) + 1) - (7x + 1)}{h} ]
Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{7x + 7h + 1 - 7x - 1}{h} ] [ f'(x) = \lim_{h \to 0} \frac{7h}{h} ]
Cancel out the ( h ) terms: [ f'(x) = \lim_{h \to 0} 7 ]
Evaluate the limit: [ f'(x) = 7 ]

Therefore, the derivative of ( f(x) = 7x + 1 ) is ( f'(x) = 7 ).

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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.