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Given ƒ(x) = 3x⁴ + 6x³ - 123x² - 126x + 1080
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 = 3, n = 4
and x is the variable we derive
ƒ''(x) = 3x⁴
ƒ''(x)( = 3 * 4)x^{(4 - 1)}
ƒ''(x) = 12x³

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Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 6, n = 3
and x is the variable we derive
ƒ''(x) = 6x³
ƒ''(x)( = 6 * 3)x^{(3 - 1)}
ƒ''(x) = 18x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -123, n = 2
and x is the variable we derive
ƒ''(x) = -123x²
ƒ''(x)( = -123 * 2)x^{(2 - 1)}
ƒ''(x) = -246x

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Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -126, n = 1
and x is the variable we derive
ƒ''(x) = -126x
ƒ''(x)( = -126 * 1)x^{(1 - 1)}
ƒ''(x) = -126

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Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 1080, n = 0
and x is the variable we derive
ƒ''(x) = 1080
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 12x³ + 18x² - 246x - 126

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 3, n = 4
and x is the variable we derive
ƒ''(x) = 3x⁴
ƒ''(x)( = 3 * 4)x^{(4 - 1)}
ƒ''(x) = 12x³

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 6, n = 3
and x is the variable we derive
ƒ''(x) = 6x³
ƒ''(x)( = 6 * 3)x^{(3 - 1)}
ƒ''(x) = 18x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -123, n = 2
and x is the variable we derive
ƒ''(x) = -123x²
ƒ''(x)( = -123 * 2)x^{(2 - 1)}
ƒ''(x) = -246x

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -126, n = 1
and x is the variable we derive
ƒ''(x) = -126x
ƒ''(x)( = -126 * 1)x^{(1 - 1)}
ƒ''(x) = -126

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 1080, n = 0
and x is the variable we derive
ƒ''(x) = 1080
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 12x³ + 18x² - 246x - 126

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 3, n = 4
and x is the variable we derive
ƒ''(x) = 3x⁴
ƒ''(x)( = 3 * 4)x^{(4 - 1)}
ƒ''(x) = 12x³

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 6, n = 3
and x is the variable we derive
ƒ''(x) = 6x³
ƒ''(x)( = 6 * 3)x^{(3 - 1)}
ƒ''(x) = 18x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -123, n = 2
and x is the variable we derive
ƒ''(x) = -123x²
ƒ''(x)( = -123 * 2)x^{(2 - 1)}
ƒ''(x) = -246x

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -126, n = 1
and x is the variable we derive
ƒ''(x) = -126x
ƒ''(x)( = -126 * 1)x^{(1 - 1)}
ƒ''(x) = -126

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 1080, n = 0
and x is the variable we derive
ƒ''(x) = 1080
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 12x³ + 18x² - 246x - 126

Start ƒ''(x)

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 3, n = 4
and x is the variable we derive
ƒ''(x) = 3x⁴
ƒ''(x)( = 3 * 4)x^{(4 - 1)}
ƒ''(x) = 12x³

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 6, n = 3
and x is the variable we derive
ƒ''(x) = 6x³
ƒ''(x)( = 6 * 3)x^{(3 - 1)}
ƒ''(x) = 18x²

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -123, n = 2
and x is the variable we derive
ƒ''(x) = -123x²
ƒ''(x)( = -123 * 2)x^{(2 - 1)}
ƒ''(x) = -246x

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = -126, n = 1
and x is the variable we derive
ƒ''(x) = -126x
ƒ''(x)( = -126 * 1)x^{(1 - 1)}
ƒ''(x) = -126

---

Use the power rule

ƒ''(x) of axⁿ = (a * n)x^{(n - 1)}
For this term, a = 1080, n = 0
and x is the variable we derive
ƒ''(x) = 1080
ƒ''(x) = 0 <--- The derivative of a constant = 0. This is part of our answer.

---

Collecting all of our derivative terms

ƒ''(x) = 12x³ + 18x² - 246x - 126

Evaluate ƒ''(0)

ƒ''(0) = 12(0)³ + 18(0)² - 246(0) - 126
ƒ''(0) = 12(0) + 18(0) - 246(0) - 126
ƒ''(0) = 0 + 0 + 0 - 126

Final Answer