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

x^n

is

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

f'(x)

with respect to

x

:

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 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.

Generate me a similar question

What is the first derivative of the function

f(x) = 5x^4 - 3x^3 + 2x^2

?

20x^3-9x^2+4x

✅Your Answer is correct

Solution by Steps

step 1

To find the first derivative of the function

f(x) = 5x^4 - 3x^3 + 2x^2

, we will use 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}(5x^4) - \frac{d}{dx}(3x^3) + \frac{d}{dx}(2x^2)

step 3

Applying the power rule to each term:

f'(x) = 5 \cdot 4x^{4-1} - 3 \cdot 3x^{3-1} + 2 \cdot 2x^{2-1}

step 4

Simplifying the expression:

f'(x) = 20x^3 - 9x^2 + 4x

Answer

The first derivative of the function

f(x) = 5x^4 - 3x^3 + 2x^2

is

f'(x) = 20x^3 - 9x^2 + 4x

.

Key Concept

Power Rule for Differentiation

Explanation

The power rule for differentiation is used to find the derivative of each term in the polynomial function, which involves multiplying the exponent by the coefficient and subtracting one from the exponent.

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