## Saturday, May 19, 2018

### Calculus/TI-84 Plus CE: Derivatives of kth Order

Calculus/TI-84 Plus CE:  Derivatives of Kth Order

Introduction

Can we find a general formula for a derivative of nth order?

d^n/dx^n f(x) = ?

The Power Function x^n

From calculus, we find that:

f(x) = x^n

d/dx x^n = n * x^(n - 1)

d^2/dx^2 x^n = n * (n – 1) * x^(n – 2)

d^3/dx^3 x^n = n * (n – 1) * (n – 2) * x^(n – 3)

d^4/dx^4 x^n = n * (n – 1) * (n – 2) * (n – 3) * x^(n – 4)

Note that for order k,

d^k/dx^k x^n = (n * (n – 1) * (n – 2) * (n – 3) * … * (n – (k – 1)) ) * x^(n – k)

d^k/dx^k x^n = (n * (n-1) * (n-2) * … * 1)/((n-k) * (n-k-1) * … * 1) * x^(n – k)

d^k/dx^k x^n = n! / (n – k)! * x^(n – k)

With the gamma function property Γ(z + 1) = z!,

d^k/dx^k x^n = Γ(n + 1)/Γ(n – k + 1) * x^(n – k)

The above formula allows us to calculate the kth derivative of x^n, even when k is not an integer.

TI-84 Plus CE Program NDERPOW

"2018-05-18 EWS"
Disp "D^K/DX^K X^N"
Input "POWER (N):",N
Input "VALUE    :",A
Input "ORDER (K):",K
If N≥K
Then
N!/(N-K)!*A^(N-K)→D
Else
0→D
End
Disp D

The program NDERPOW calculates the numerical derivative of d^k/dx^k x^n.  For this particular program, k must be an integer since non-integers are not accepted on the TI-84 Plus’ factorial function.

The Exponential Function e^(a*x), where a is a constant

f(x) = e^(a*x)

d/dx e^(a*x) = a * e^(a*x)

d^2/dx^2 e^(a*x) = a^2 * e^(a*x)

d^3/dx^3 e^(a*x) = a^3 * e^(a*x)

d^4/dx^4 e^(a*x) = a^4 * e^(a*x)

With the order k…

d^k/dx^k e^(a*x) = a^k * e^(a*x)

Just like the last case, k does not have be an integer.

TI-84 Plus CE Program NDEREXP

"2015-05-18 EWS"
Disp "D^K/DX^K e^(A*X)"
Input "COEFF (A):",A
Input "VALUE    :",X
Input "ORDER (K):",K
A^K*e^(A*X)→D
Disp D

The program NDEREXP calculates the numerical derivative of d^k/dx^k e^(a*x).

The Sine Function sin(a*x) and the Cosine Function cos(a*x)

We are working radian angle measure.

f(x) = sin(a*x)

d/dx sin(a*x) = a * cos(a*x)

d^2/dx^2 sin(a*x) = -a^2 * sin(a*x)

d^3/dx^3 sin(a*x) = -a^3 * cos(a*x)

d^4/dx^4 sin(a*x) = a^4 * sin(a*x)

Notice a pattern, alternating between sin and cos.  To the kth order (k is an integer),

d^k/dx^k sin(a*x) =

(-1)^int(k/2) * a^k * cos(a*x), when k is odd

(-1)^int(k/2) * a^k * sin(a*x), when k is even

If we put the piecewise function into one statement:

d^k/dx^k sin(a*x) = (-1)^int(k/2) * a^k * ( 2*frac(k/2)*cos(a*x) + 2*frac((k+1)/2)*sin(a*x) )

Note that:

2 * frac(k/2) = 1 for all odd integers k, 0 for all even integers k

Likewise, 2 * frac((k + 1)/2) = 0 for all odd integers k, 1 for all even integers k

And, (-1)^int(k/2) produces a pattern of 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, …

TI-84 Plus CE Program NDERSIN

"2018-05-19 EWS"
Disp "D^K/DX^K sin(A*X)"
Input "COEFF (A):",A
Input "VALUE    :",X
Input "ORDER (K):",K
(­1)^iPart(K/2)*A^K*(2*fPart(K/2)*cos(A*X)+2*fPart((K+1)/2)*sin(A*X))→D
Disp D

Similarly,

f(x) = cos(a*x)

d/dx cos(a*x) = -a * sin(a*x)

d^2/dx^2 cos(a*x) = -a^2 * cos(a*x)

d^3/dx^3 cos(a*x) = a^3 * sin(a*x)

d^4/dx^4 cos(a*x) = a^4 * cos(a*x)

Likewise:

Notice a pattern, alternating between sin and cos.  To the kth order (k is an integer),

d^k/dx^k sin(a*x) =

(-1)^int(k/2 + 1/2) * a^k * sin(a*x), when k is odd

(-1)^int(k/2 + 1/2) * a^k * cos(a*x), when k is even

If we put the piecewise function into one statement:

d^k/dx^k cos(a*x) = (-1)^int(k/2 + 1/2) * a^k * ( 2*frac(k/2)*sin(a*x) + 2*frac((k+1)/2)*cos(a*x) )

TI-84 Plus CE Program NDERCOS

"2018-05-19 EWS"
Disp "D^K/DX^K cos(A*X)"
Input "COEFF (A):",A
Input "VALUE    :",X
Input "ORDER (K):",K
(­1)^iPart((K+1)/2)*A^K*(2*fPart(K/2)*sin(A*X)+2*fPart((K+1)/2)*cos(A*X))→D
Disp D

NDERSIN and NDERCOS are numeric derivatives for sine and cosine, respectively.  Note for NDERSIN and NDERCOS, the calculator is set to Radian mode, and K (order) should be an integer.

Eddie

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