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Determine the second derivative of the function

f(x) = 3x^3 - 2x^2 + 4x

Jan 3, 2024

Determine the second derivative of the function

f(x) = 3x^3 - 2x^2 + 4x

Here is a Step-by-Step solution: #step 1#

To find the second derivative of the function

f(x) = 3x^3 - 2x^2 + 4x

, we first need to find the first derivative

f'(x)

. The first derivative represents the rate of change or the slope of the function

f(x)

#step 2#

Differentiating the function

f(x)

with respect to

x

to find the first derivative

f'(x)

f'(x) = \frac{d}{dx}(3x^3) - \frac{d}{dx}(2x^2) + \frac{d}{dx}(4x)

#step 3#

Applying the power rule of differentiation, which states that

\frac{d}{dx}(x^n) = nx^{n-1}

, to each term:

f'(x) = 3 \cdot 3x^{3-1} - 2 \cdot 2x^{2-1} + 4 \cdot 1x^{1-1}

#step 4#

Simplifying the expression for the first derivative:

f'(x) = 9x^2 - 4x + 4

#step 5#

Now, to find the second derivative

f''(x)

, we differentiate the first derivative

f'(x)

with respect to

x

#step 6#

Differentiating the first derivative

f'(x)

with respect to

x

to find the second derivative

f''(x)

f''(x) = \frac{d}{dx}(9x^2) - \frac{d}{dx}(4x) + \frac{d}{dx}(4)

#step 7#

Applying the power rule of differentiation again to each term:

f''(x) = 9 \cdot 2x^{2-1} - 4 \cdot 1x^{1-1} + 0

#step 8#

Simplifying the expression for the second derivative:

f''(x) = 18x - 4

Answer

The second derivative of the function

f(x) = 3x^3 - 2x^2 + 4x

f''(x) = 18x - 4

Key Concept

The second derivative of a function is found by differentiating the first derivative of the function.

Explanation

The second derivative

f''(x)

represents the rate of change of the slope of the function

f(x)

, which in this case is

18x - 4

. This can indicate the concavity of the function and acceleration if the function represents a position with respect to time.