Calculating the Rate of Change | Finding f'(2) for the Function f(x) = x^3 + 3x^2 – 4

Let f be the function given by f(x)=x^3+3x^2−4. What is the value of f′(2) ?

To find the value of f'(2), we first need to find the derivative of the function f(x)

To find the value of f'(2), we first need to find the derivative of the function f(x). The derivative of a function represents the rate at which the function is changing at a particular value of x.

The given function is f(x) = x^3 + 3x^2 – 4. To find its derivative, we can apply the power rule, which states that the derivative of x^n is n*x^(n-1).

Taking the derivative of each term separately, we have:
f'(x) = d/dx (x^3) + d/dx (3x^2) – d/dx (4)

Using the power rule, the derivative of each term is:
f'(x) = 3x^2 + 6x – 0

Now, we have the derivative function f'(x) = 3x^2 + 6x.

To find the value of f'(2), we substitute x = 2 into the derivative function:
f'(2) = 3(2)^2 + 6(2)
f'(2) = 3(4) + 12
f'(2) = 12 + 12
f'(2) = 24

Therefore, the value of f'(2) is 24.

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