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Determine the second derivative of the function

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

Dec 25, 2023

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 use the power rule for differentiation, which states that the derivative of

x^n

nx^{n-1}

step 2

Differentiating the given function

f(x)

with respect to

x

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

step 3

To find the second derivative of the function

f(x)

, we differentiate

f'(x)

with respect to

x

using the power rule again

step 4

Differentiating the first derivative

f'(x)

with respect to

x

f''(x) = 18x - 4

Answer

The second derivative of the function

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

f''(x) = 18x - 4

Key Concept

Power Rule for Differentiation

Explanation

The power rule is used to differentiate terms of the form

x^n

, resulting in

nx^{n-1}

. Applying this rule twice gives us the second derivative of the function.