Given ƒ(x) = 2x
3 - 4x
2 - 22x + 24
Determine the 2nd derivative ƒ''(x)
Start ƒ''(x)
Use the power rule
ƒ''(x) of ax
n = (a * n)x
(n - 1)For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x
3ƒ''(x)( = 2 * 3)x
(3 - 1)ƒ''(x) = 6x
2
Use the power rule
ƒ''(x) of ax
n = (a * n)x
(n - 1)For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x
2ƒ''(x)( = -4 * 2)x
(2 - 1)ƒ''(x) = -8x
Use the power rule
ƒ''(x) of ax
n = (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 ax
n = (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 ax
n = (a * n)x
(n - 1)For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x
3ƒ''(x)( = 2 * 3)x
(3 - 1)ƒ''(x) = 6x
2
Use the power rule
ƒ''(x) of ax
n = (a * n)x
(n - 1)For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x
2ƒ''(x)( = -4 * 2)x
(2 - 1)ƒ''(x) = -8x
Use the power rule
ƒ''(x) of ax
n = (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 ax
n = (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 ax
n = (a * n)x
(n - 1)For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x
3ƒ''(x)( = 2 * 3)x
(3 - 1)ƒ''(x) = 6x
2
Use the power rule
ƒ''(x) of ax
n = (a * n)x
(n - 1)For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x
2ƒ''(x)( = -4 * 2)x
(2 - 1)ƒ''(x) = -8x
Use the power rule
ƒ''(x) of ax
n = (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 ax
n = (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 ax
n = (a * n)x
(n - 1)For this term, a = 2, n = 3
and x is the variable we derive
ƒ''(x) = 2x
3ƒ''(x)( = 2 * 3)x
(3 - 1)ƒ''(x) = 6x
2
Use the power rule
ƒ''(x) of ax
n = (a * n)x
(n - 1)For this term, a = -4, n = 2
and x is the variable we derive
ƒ''(x) = -4x
2ƒ''(x)( = -4 * 2)x
(2 - 1)ƒ''(x) = -8x
Use the power rule
ƒ''(x) of ax
n = (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 ax
n = (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) - 22
ƒ''(0) = 6(0) - 8(0) - 22
ƒ''(0) = 0 + 0 - 22
Final Answer
ƒ''(0) = -22
You have 2 free calculationss remaining
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) = x
n, 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
Example calculations for the Functions-Derivatives-Integrals Calculator
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