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?
The definition is:
So:
that is the derivative of the function.
By signing up, you agree to our Terms of Service and Privacy Policy
To calculate the derivative of ( f(x) = 3x^3 + 9x + 4 ) using the limit definition of the derivative, 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) = 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} ]

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) ]
 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 ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 Using the limit definition, how do you find the derivative of #f (x) = sqrt (x−3)#?
 How do you solve this problem step by step?
 What is the equation of the tangent line of #f(x)=sqrt(x^24x+7)/(x1)# at #x=3#?
 How do you find the slope of the tangent to the curve #y = 3 + 4x^2  2x^3# where x = a?
 How do you find the average rate of change of #f(x) = sec(x)# from #x=0# to #x=pi/4#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7