Question

Use the definition of a derivative to find f '(x) and f ''(x).
f(x) = 3x² + 9x+2

Answer 1

The first derivative of the function f(x) = 3x² + 9x + 2, found using the power rule, is f'(x) = 6x + 9, and the second derivative, by differentiating the first, is f''(x) = 6. These derivatives represent the velocity and acceleration if f(x) is a position function.

Step-by-step explanation:

To find the first derivative (f'(x)) of the function f(x) = 3x² + 9x + 2 using the definition of a derivative, we will apply the limit process. However, for polynomial functions such as this one, it's more common to directly apply differentiation rules. We use the power rule for differentiation, which states that the derivative of xⁿ with respect to x is n*xⁿ⁻¹. Applying this rule to each term gives:

• The derivative of 3x² is 2*3x²⁻¹ = 6x.
• The derivative of 9x is 9, since the derivative of x is 1.
• The derivative of the constant 2 is 0.

So, f'(x) = 6x + 9.

Next, we find the second derivative (f''(x)) by differentiating f'(x). Again applying the power rule:

• The derivative of 6x is 6, since the derivative of x is 1.
• The derivative of a constant (9) is 0.

So, f''(x) = 6.

Therefore, the first derivative represents the velocity function if f(x) is a position function, while the second derivative represents the acceleration function.