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How do you find the derivative of the function #f(x)=x^2-3x# using #f(x+h)-f(x)/h#?

Answer 1

Ezekiel Alexander

Know that:

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

#=lim_(hrarr0)((x+h)^2-3(x+h)-(x^2-3x))/h#

#=lim_(hrarr0)(x^2+2xh+h^2-3x-3h-x^2+3x)/(h)#

#=lim_(hrarr0)(h^2+2xh-3h)/(h)#

#=lim_(hrarr0)(h(h+2x-3))/(h)#

#=lim_(hrarr0)h+2x-3#

#=2x-3#

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Answer 2

Adam Adkins

To find the derivative of the function ( f(x) = x^2 - 3x ) using the formula ( \frac{f(x+h) - f(x)}{h} ), you first substitute ( f(x+h) ) and ( f(x) ) into the formula.

( f(x+h) = (x+h)^2 - 3(x+h) = x^2 + 2xh + h^2 - 3x - 3h )

Substitute into the formula:

( \frac{(x^2 + 2xh + h^2 - 3x - 3h) - (x^2 - 3x)}{h} )

Now simplify the expression:

( \frac{x^2 + 2xh + h^2 - 3x - 3h - x^2 + 3x}{h} )

Now cancel out like terms:

( \frac{2xh + h^2 - 3h}{h} )

Now factor out an ( h ):

( \frac{h(2x + h - 3)}{h} )

Now cancel out ( h ) from numerator and denominator:

( 2x + h - 3 )

As ( h ) approaches 0, the ( h ) term disappears, leaving:

( \boxed{2x - 3} )

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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.