CALCU-Determine the second derivative of the funct

Determine the second derivative of the function

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

.

Solution by Steps

step 1

To find the first derivative of the function

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

, we apply the power rule for differentiation, which states that the derivative of

x^n

is

nx^{n-1}

step 2

Differentiating each term of the function separately:

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

step 3

Applying the power rule to each term:

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

step 4

Simplifying the first derivative:

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

step 5

To find the second derivative, we differentiate

f'(x)

with respect to

x

again:

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

step 6

Applying the power rule to each term of the first derivative:

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

step 7

Simplifying the second derivative:

f''(x) = 18x - 4

Answer

The second derivative of the function

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

is

f''(x) = 18x - 4

.

Key Concept

Power Rule for Differentiation

Explanation

The power rule for differentiation is used to find the derivative of a function with respect to

x

, which involves reducing the exponent by one and multiplying by the original exponent. This process is repeated to find the second derivative.

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How to calculate the first derivative of the function

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

?

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