Functions-Derivatives-Integrals Calculator

Enter your expression below:

ƒ()
ƒ'()
n =

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Evaluate the function
ƒ(n) = n⁵²
at ƒ(0)
Given ƒ(n) = n⁵²
Determine the derivative ∫ƒ(n)
Given ƒ(n) = n⁵²
Determine the 2nd derivative ∫ƒ(n)
Given ƒ(n) = n⁵²dn
Determine the integral ∫ƒ(n)
Go through and integrate each term
Given the equation ƒ(n) = n⁵²
use Simpsons Rule with n =
over the interval [0,1]

Substitute each n with 0

ƒ(0) = (0)⁵²
ƒ(0) = (0)
ƒ(0) = 0

Start ƒ'(n)

Use the power rule

ƒ'(n) of anⁿ = (a * n)n^{(n - 1)}
For this term, a = 1, n = 52
and n is the variable we derive
ƒ'(n) = n⁵²
ƒ'(n)( = 1 * 52)n^{(52 - 1)}
ƒ'(n) = 52n⁵¹

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Collecting all of our derivative terms

ƒ'(n) = 52n⁵¹

Start ƒ''(n)

Use the power rule

ƒ''(n) of anⁿ = (a * n)n^{(n - 1)}
For this term, a = 52, n = 51
and n is the variable we derive
ƒ''(n) = 52n⁵¹
ƒ''(n)( = 52 * 51)n^{(51 - 1)}
ƒ''(n) = 2652n⁵⁰

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Collecting all of our derivative terms

ƒ''(n) = 2652n⁵⁰

Start ƒ⁽³⁾(n)

Use the power rule

ƒ⁽³⁾(n) of anⁿ = (a * n)n^{(n - 1)}
For this term, a = 2652, n = 50
and n is the variable we derive
ƒ⁽³⁾(n) = 2652n⁵⁰
ƒ⁽³⁾(n)( = 2652 * 50)n^{(50 - 1)}
ƒ⁽³⁾(n) = 132600n⁴⁹

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Collecting all of our derivative terms

ƒ⁽³⁾(n) = 132600n⁴⁹

Start ƒ⁽⁴⁾(n)

Use the power rule

ƒ⁽⁴⁾(n) of anⁿ = (a * n)n^{(n - 1)}
For this term, a = 132600, n = 49
and n is the variable we derive
ƒ⁽⁴⁾(n) = 132600n⁴⁹
ƒ⁽⁴⁾(n)( = 132600 * 49)n^{(49 - 1)}
ƒ⁽⁴⁾(n) = 6497400n⁴⁸

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Collecting all of our derivative terms

ƒ⁽⁴⁾(n) = 6497400n⁴⁸

Evaluate ƒ⁽⁴⁾(0)

ƒ⁽⁴⁾(0) = 6497400(0)⁴⁸
ƒ⁽⁴⁾(0) = 6497400(0)
ƒ⁽⁴⁾(0) = 0
Given ƒ(n) = n⁵²dn
Determine the integral ∫ƒ(n)
Go through and integrate each term

Integrate term 1

ƒ(n) = 6497400n⁴⁸

Use the power rule

∫ƒ(n) of the expression anⁿ

an(n + 1)
n + 1

= 6497400, n = 48
and n is the variable we integrate

∫ƒ(n)  =	6497400n(48 + 1)
	48 + 1

∫ƒ(n)  =	6497400n49
	49

Simplify our fraction.
Divide top and bottom by 49
∫ƒ(n) = 132600n⁴⁹

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Collecting all of our integrated terms we get:

∫ƒ(n) = 6497400n⁴⁹132600n⁴⁹

Evaluate ∫ƒ(n) on the interval [0,1]

The value of the integral over an interval is ∫ƒ(1) - ∫ƒ(0)

Evaluate ∫ƒ(1)

∫ƒ(1) = 6497400(1)⁴⁹132600(1)⁴⁹
∫ƒ(1) = 6497400(1)132600(1)
∫ƒ(1) = 132600
∫ƒ(1) = 132600

Evaluate ∫ƒ(0)

∫ƒ(0) = 6497400(0)⁴⁹132600(0)⁴⁹
∫ƒ(0) = 6497400(0)132600(0)
∫ƒ(0) = 0
∫ƒ(0) = 0

Determine our answer

∫ƒ(n) on the interval [0,1] = ∫ƒ(1) - ∫ƒ(0)
∫ƒ(n) on the interval [0,1] = 132600 - 0

Final Answer