Question

differential of y=(1+cos^2(3x))^2

I got (1+cos^2(3x))(cos3x)(-sin3x)
but my answer key says (-3cos3xsin3x)/(root of (1+cos^2(3x)))

Answer 1

Differential of $y = (1 + \cos^2(3x))^2$

Solution:

If your task is like in the statement, then

$$((1 + (\cos 3x)^2)^2)' = -12 \sin(3x) \cos(3x)(1 + (\cos 3x)^2)$$

But if in the statement there is an error and your task is $y = \sqrt{1 + (\cos 3x)^2}$, then

$$y' = \frac{(1 + (\cos 3x)^2)'}{2\sqrt{1 + (\cos 3x)^2}} = \frac{2 \cos 3x (\cos 3x)'}{2\sqrt{1 + (\cos 3x)^2}} = \frac{-3 \cos 3x \sin 3x}{\sqrt{1 + (\cos 3x)^2}}$$

And this answer is like in your answer key. So, possibly, there is an error in the statement.