Functions-Derivatives-Integrals Calculator
Publish date: 2024-07-24
Given ƒ(x) = 2x3 - 4x2 - 22x + 24Determine 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 - 22Start 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 - 22Start 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 - 22Start 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 - 22Evaluate ƒ''(0)
ƒ''(0) = 6(0)2 - 8(0) - 22ƒ''(0) = 6(0) - 8(0) - 22
ƒ''(0) = 0 + 0 - 22
Final Answer
ƒ''(0) = -22
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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) 1st Derivative ƒ‘(x) The derivative of your expression will also be evaluated at a point, i.e., ƒ‘(1)
3) 2nd 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?
derivativerate at which the value y of the function changes with respect to the change of the variable xexponentThe power to raise a numberfunctionrelation between a set of inputs and permissible outputsƒ(x)functions-derivatives-integralsintegrala mathematical object that can be interpreted as an area or a generalization of areapointan exact location in the space, and has no length, width, or thicknesspolynomialan expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).powerhow many times to use the number in a multiplication
Example calculations for the Functions-Derivatives-Integrals Calculator
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