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Derivatives of Polynomial Functions

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Derivatives of Polynomial Functions - Lesson Summary

Polynomial of degree n

f(x) = anxn + an-1xn-1 + .... + a2x2 + a1x + a0 where, ai = Real number,∀ i ∈ Z and an an ≠ 0.

Theorem: Derivative of f(x) = xn is nxn-1 for any positive integer n.


Derivative of the function using the first principle:

 f ' (x) =   lim    h → 0 f(x+h)-f(x) h

f ' (x) = lim h → 0 (x + h) n - x n h

From the binomial theorem,
(x+h)n = (nC0xn + nC1xn-1 h + nC2xn-2 h2 + .... + nCn-1 xhn-1 + nCn hn)

⇒ (x+h)n = (xn + nxn-1 h + n(n-1)/2 xn-2 h2 + .... + n xhn-1 + hn)

⇒ (x+h)n - xn = h( nxn-1 + n(n-1)/2 xn-2 h + .... + n xhn-2 + hn-1)

Using the above in the derivative formula for f(x)

f ' (x) = lim        h → 0   (h( nxn-1 + n(n-1)/2 xn-2 h + .... + n xhn-2 + hn-1))/h

⇒ fI(x) = nxn-1 + 0 + ... + 0

⇒ fI(x) = nxn-1

d/dx xn = nxn-1

Derivative of xn using product rule

xn = x. xn-1

d/dx(xn ) = d/dx( x. xn-1)

= xn-1. d/dx(x) + x. d/dx(xn-1)

Since d/dx(xn ) = xn-1 + x. d/dx(xn-1)

Since d/dx(xn ) = xn-1 + x.[xn-2 + x d/dx(xn-2)]

= xn-1 + xn-1 + x2 d/dx(xn-2)

= 2xn-1 + x2 d/dx(xn-2)

Since d/dx (xn) = 2xn-1+x2[xn-3 + x.d/dx(xn-3)]

= 3xn-1 + x3. d/dx(xn-3)

⇒ d/dx (xn) = nxn-1 + xn. d/dx (xn-n)

⇒ d/dx (xn) = nxn-1 + xn. d/dx (1)

⇒ d/dx (xn) = nxn-1 + xn. 0

∴ d/dx(xn) = nxn-1

The derivative of xn is applicable even when n is real.

Ex: Derivative of f(x) = 50x4

fI(x) = 50 d/dx (x4) = 50.4.x4-1 = 200 x3.

Derivatives of Polynomial Functions

f(x) = anxn + an-1xn-1 + .... + a2x2 + a1x + a0 where, ai = Real number,∀ i ∈ Z and an an ≠ 0.

d/dx f(x) = nanxn-1 + (n-1)an-1xn-2 + .... + 2a2x + a1

Proof:

d/dx f(x) = d/dx[anxn + an-1xn-1 + .... + a2x2 + a1x + a0]

d/dx f(x) = d/dx(anxn) + d/dx(an-1xn-1) + .... + d/dx(a2x2) + d/dx(a1x) + d/dx(a0)

[Since d/dx[f(x) + g(x)] = d/dx f(x) + d/dx g(x)]

d/dx f(x) = an d/dx(xn) + an-1 d/dx(xn-1) + .... + a2 d/dx(x2) + a1 d/dx(x) + d/dx(a0)

Derivative of each term:

d/dx f(x) = nan xn-1 + (n-1)an-1 xn-2 + .... + 2a2 x + a1

[Since d/dx(kf(x)) = kd/dx(f(x)) and d/dx xn = nxn-1]

g(x) = 5x4 + 3x2 - 2x + 1

Since the function is a polynomial function, the theorem can be directly applied

d/dx[g(x)] = 5d/dx (x4) + 3d/dx (x2) - 2d/dx(x) + d/dx(1)

= 5.4x4-1 + 3.2x2-1 - 2. x1-1 + 0

= 20x3 + 6x -2.

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