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ƒ()
ƒ'()
n =

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Given ƒ(x) = 2x³ - 4x² - 22x + 24
Determine the 2nd derivative ƒ''(x)

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x³
ƒ''(x)( = 2 * 3)x^{(3 - 1)}
ƒ''(x) = 6x²

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Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x²
ƒ''(x)( = -4 * 2)x^{(2 - 1)}
ƒ''(x) = -8x

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Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -22, n = 1
and x is the variable we derive
ƒ''(x) = -22x
ƒ''(x)( = -22 * 1)x^{(1 - 1)}
ƒ''(x) = -22

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Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 24, n = 0
and x is the variable we derive
ƒ''(x) = 24
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

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Collecting all of our derivative terms

ƒ''(x) = 6x² - 8x - 22

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x³
ƒ''(x)( = 2 * 3)x^{(3 - 1)}
ƒ''(x) = 6x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x²
ƒ''(x)( = -4 * 2)x^{(2 - 1)}
ƒ''(x) = -8x

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -22, n = 1
and x is the variable we derive
ƒ''(x) = -22x
ƒ''(x)( = -22 * 1)x^{(1 - 1)}
ƒ''(x) = -22

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 24, n = 0
and x is the variable we derive
ƒ''(x) = 24
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 6x² - 8x - 22

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x³
ƒ''(x)( = 2 * 3)x^{(3 - 1)}
ƒ''(x) = 6x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x²
ƒ''(x)( = -4 * 2)x^{(2 - 1)}
ƒ''(x) = -8x

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -22, n = 1
and x is the variable we derive
ƒ''(x) = -22x
ƒ''(x)( = -22 * 1)x^{(1 - 1)}
ƒ''(x) = -22

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 24, n = 0
and x is the variable we derive
ƒ''(x) = 24
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 6x² - 8x - 22

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x³
ƒ''(x)( = 2 * 3)x^{(3 - 1)}
ƒ''(x) = 6x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x²
ƒ''(x)( = -4 * 2)x^{(2 - 1)}
ƒ''(x) = -8x

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -22, n = 1
and x is the variable we derive
ƒ''(x) = -22x
ƒ''(x)( = -22 * 1)x^{(1 - 1)}
ƒ''(x) = -22

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 24, n = 0
and x is the variable we derive
ƒ''(x) = 24
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 6x² - 8x - 22

Evaluate ƒ''(0)

ƒ''(0) = 6(0)² - 8(0) - 22
ƒ''(0) = 6(0) - 8(0) - 22
ƒ''(0) = 0 + 0 - 22

Answer

↓Steps Explained:↓

Final Answer

ƒ''(0) = -22

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