# Find the value of the derivative of the function f (x) at the point x0 = 2 if f (x) = 3x ^ 3-x ^ 2-7.

July 28, 2021 | education

| Find the derivative of this function: f (x) = 3x ^ 3 – x ^ 2 – 7.

Using the formulas:

(x ^ n) ’= n * x ^ (n-1) (derivative of the basic elementary function).

(c) ‘= 0, where c – const (derivative of the basic elementary function).

(c * u) ’= c * u’, where c is const (basic rule of differentiation).

(u + v) ’= u’ + v ’(basic rule of differentiation).

Thus, the derivative of our function will be as follows:

f (x) ‘= (3x ^ 3 – x ^ 2 – 7)’ = (3x ^ 3) ‘- (x ^ 2)’ – (7) ‘= 3 * 3 * x ^ (3 – 1) – 2 * x ^ (2 – 1) – 0 = 9x ^ 2 – 2x.

We calculate the value of the derivative at the point x0 = 2:

f (x) ‘(2) = 9 * 2 ^ 2 – 2 * 2 = 9 * 4 – 4 = 36 – 4 = 32.

Answer: f (x) ‘= 9x ^ 2 – 2x, and f (x)’ (2) = 32.

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