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Using the limit definition, how do you differentiate #f(x) = 7x + 8#?

Answer 1

Isaiah Allen

See the explanation section below.

#f(x) = 7x+8#

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

# = lim_(hrarr0)([7(x+h)+8]-[7x+8])/h#

# = lim_(hrarr0)(7x+7h+8 - 7x - 8)/h#

# = lim_(hrarr0)(7h)/h#

# = lim_(hrarr0)7#

# = 7#

So, #f'(x) = 7#

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Answer 2

Elijah Abram

To differentiate the function ( f(x) = 7x + 8 ) using the limit definition, you follow these steps:

Start with 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 + 8 ) into the limit definition: [ f'(x) = \lim_{h \to 0} \frac{(7(x + h) + 8) - (7x + 8)}{h} ]
Simplify the expression inside the limit: [ f'(x) = \lim_{h \to 0} \frac{7x + 7h + 8 - 7x - 8}{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 + 8 ) with respect to ( x ) 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.