Let #f(x) = -3 x^3 + 9 x + 4#, how do you use the limit definition of the derivative to calculate the derivative of f?

Answer 1

The definition is:

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

So:

#lim_(hrarr0)(-3(x+h)^3+9(x+h)+4-(-3x^3+9x+4))/h=#
#=lim_(hrarr0)(-3(x^3+3x^2h+3xh^2+h^3)+9x+9h+4+3x^3-9x-4)/h=#
#=lim_(hrarr0)(-3x^3-9x^2h-9xh^2-3h^3+9x+9h+4+3x^3-9x-4)/h=#
#=lim_(hrarr0)(-9x^2h-9xh^2-3h^3+9h)/h=#
#=lim_(hrarr0)(-9x^2-9xh-3h^2+9)=-9x^2+9#,

that is the derivative of the function.

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Answer 2

To calculate the derivative of ( f(x) = -3x^3 + 9x + 4 ) using the limit definition of the derivative, follow these steps:

  1. Write down the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

  2. Substitute the function ( f(x) = -3x^3 + 9x + 4 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{-3(x+h)^3 + 9(x+h) + 4 - (-3x^3 + 9x + 4)}{h} ]

  3. Expand and simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{-3(x^3 + 3x^2h + 3xh^2 + h^3) + 9x + 9h + 4 + 3x^3 - 9x - 4}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-3x^3 - 9x^2h - 9xh^2 - 3h^3 + 9x + 9h + 4 + 3x^3 - 9x - 4}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-9x^2h - 9xh^2 - 3h^3 + 9h}{h} ]

[ f'(x) = \lim_{h \to 0} (-9x^2 - 9xh - 3h^2 + 9) ]

  1. Evaluate the limit as ( h ) approaches 0: [ f'(x) = -9x^2 + 9 ]

Therefore, the derivative of ( f(x) = -3x^3 + 9x + 4 ) is ( f'(x) = -9x^2 + 9 ).

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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.

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.

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.

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.

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