The first few derivatives of y=2sin(3t−1) with respect to t are:
dtdydt2d2ydt3d3ydt4d4ydt5d5y=6cos(3t−1)=−18sin(3t−1)=−54cos(3t−1)=162sin(3t−1)=486cos(3t−1)
Given a positive integer n, let r denote the remainder when n is divided by 4.
The nth derivative of 2sin(3t−1) with respect to t is given by the formula:
dtndn[2sin(3t−1)]=⎩⎨⎧2⋅3nsin(3t−1)(−1)2⋅3ncos(3t−1)2⋅3nsin(3t−1)(−1)2⋅3ncos(3t−1) when r=0 when r=1 when r=2 when r=3
In general, the nnth derivative of asin(bx+c) is given by the formula:
dxndn[asin(bx+c)]=⎩⎨⎧abnsin(bx+c)abncos(bx+c)−abnsin(bx+c)−abncos(bx+c) when r=0 when r=1 when r=2 when r=3
In the case of 2sin(3t−1), a=2, b=3 and c=−1.