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How do you find the derivative of #f(x)=-5x# using the limit process?

Answer 1

Isaiah Allen

see below

the definition

#(dy)/(dx)=Lim_(hrarr0)((f(x+h)-f(x))/h)#

#f(x)=-5x#

#(dy)/(dx)=Lim_(hrarr0)((-5(x+h)- (-5x))/h)#

#(dy)/(dx)=Lim_(hrarr0)((cancel(-5x)-5hcancel(+5x))/h)#

#(dy)/(dx)=Lim_(hrarr0)((-5cancel(h))/cancel(h))#

#(dy)/(dx)=Lim_(hrarr0)(-5)=-5#

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Answer 2

Elijah Abram

To find the derivative of ( f(x) = -5x ) using the limit process, you apply the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute the function ( f(x) = -5x ) into the formula:

[ f'(x) = \lim_{h \to 0} \frac{-5(x + h) - (-5x)}{h} ]

Simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{-5x - 5h + 5x}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-5h}{h} ]

Cancel out the ( h ) terms:

[ f'(x) = \lim_{h \to 0} -5 ]

As ( h ) approaches 0, the derivative is constant:

[ f'(x) = -5 ]

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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.