Differentiation

n^th Derivative

We already learned about second derivatives, which is the derivative of a derivative, but what happens when we keep taking the derivative of a function?

n^th Derivative

The nth derivative of f is

f(n)(x)=f′′′′…′′n times(x)🚫

dxndny=dxd(dxd(…dxd(y)))n times🚫

For example, look at the notation in use below:

f(x)=x5 dxdf′(x)=5x4dxdf(2)(x)=20x3dxdf(3)(x)=60x2dxdf(4)(x)=120xdxdf(5)(x)=120.

The IB often tests higher order derivatives by giving nth power proofs, which involve induction.

Discussion

Prove that iff(x)=ekx, then f(n)(x)=knekx by induction.

Base case (n = 0):

f(0)(x)=f(x)=ekx=k0ekx

Inductive step: assume for some n≥0 that

f(n)(x)=knekx

Then

f(n+1)(x)=dxd(f(n)(x))=dxd(knekx)=kndxd(ekx)=kn⋅kekx=kn+1ekx

By the principle of mathematical induction, f(n)(x)=knekx for all n∈N0.

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