Question

If y = e* sin^2x , find yn
x ,i. e. the nth derivative of y​

Answer 1

Answer:

Let’s do it in the pattern recognition method. The main mechanism behind this is sinx=cos(π2−x) and sin(2x)=2sinxcosx .

y1=ddxsin2(x)

Let u=sin(x)

Then by the chain rule :

y1=dduu2.ddxsinx

y1=2sinxcosx

y1=sin(2x)

Then let’s go for the second derivative :

y2=ddxsin(2x)

Then by the chain rule :

Let v=2x

y2=ddvsinvddx2x

y2=2cos(2x)

y2=2sin(π2−2x)

Then for the third derivative :

y3=ddx2sin(π2−2)

Then again, by the chain rule :

Let w=π2−2x

y3=2ddwsinwddx(π2−2x)

y3=−22cos(π2−2x)

y3=22cos(2x−π2)

y3=22sin(2π2−2x)

Then for the fourth derivative too :

y4=ddx22sin(2π2−2x)

Then again using the chain rule :

Let t=2π2−2x

y4=22ddtsintddx2π2−2x

y4=−23cos(2π2−2x)

y4=23cos(2x−2π2

y4=23sin[π2−(2x−2π2)]

y4=23sin(3π2−2x)

Thus, the pattern is very clear, the n-th derivative will be :

yn=2(n−1)sin((n−1)π2−2x)

That is it!

With regards.