Functions-Derivatives-Integrals Calculator

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

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

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

Final Answer

ƒ''(0) = -22

You have 1 free calculations remaining

What is the Answer?

ƒ''(0) = -22

How does the Functions-Derivatives-Integrals Calculator work?

Free Functions-Derivatives-Integrals Calculator - Given a polynomial expression, this calculator evaluates the following items:
1) Functions ƒ(x).  Your expression will also be evaluated at a point, i.e., ƒ(1)
2) 1^st Derivative ƒ‘(x)  The derivative of your expression will also be evaluated at a point, i.e., ƒ‘(1)
3) 2^nd Derivative ƒ‘‘(x)  The second derivative of your expression will be also evaluated at a point, i.e., ƒ‘‘(1)
4) Integrals ∫ƒ(x)  The integral of your expression will also be evaluated on an interval, i.e., [0,1]
5) Using Simpsons Rule, the calculator will estimate the value of ≈ ∫ƒ(x) over an interval, i.e., [0,1]
This calculator has 7 inputs.

What 1 formula is used for the Functions-Derivatives-Integrals Calculator?

Power Rule: f(x) = xⁿ, f‘(x) = nx^{(n - 1)}

For more math formulas, check out our Formula Dossier

What 8 concepts are covered in the Functions-Derivatives-Integrals Calculator?

derivative: rate at which the value y of the function changes with respect to the change of the variable x
exponent: The power to raise a number
function: relation between a set of inputs and permissible outputs
ƒ(x)
functions-derivatives-integrals
integral: a mathematical object that can be interpreted as an area or a generalization of area
point: an exact location in the space, and has no length, width, or thickness
polynomial: an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
power: how many times to use the number in a multiplication