Here is our function:
#f(x) = ln(sin x)^2 - 3x ln(sin x) + x^2 ln(cos^2 x^2)#
Taking the derivative, we should have three smaller parts:
#(df)/(dx) = (d/dx ln(sin x)^2) - (d/dx 3x ln(sin x)) + (d/dx x^2 ln(cos^2 x^2))#
Let's label them:
#g(x) = ln(sin x)^2#
#h(x) = 3x ln(sin x)#
#p(x) = x^2 ln(cos^2 x^2)#
So it becomes:
#(df)/(dx) = (dg)/(dx) - (dh)/(dx) + (dp)/(dx)#
Let's find the derivatives one-by-one, starting from #g(x)#. Let's... break this function composition down.
#g_1 (x) = sin(x)#
#g_2 (x) = ln(x)#
#g_3 (x) = x^2#
So #g(x) = g_3 (g_2 (g_1 (x)))#. Let's take the derivative, one by one, using the chain rule:
#dg_1 = d(sin(x)) = cos(x) dx#
#(dg_1)/(dx) = cos(x)#
#dg_2 = d(ln(g_1)) = (g_1)^-1 dg_1 = (sin(x))^-1 cos(x) dx#
#(dg_2)/(dx) = csc(x) cos(x)#
#dg_3 = d((g_2)^2) = 2g_2 dg_2 = 2 ln(g_1) csc(x) cos(x) dx#
#= 2 ln(sin(x)) csc(x) cos(x) dx#
#(dg_3)/(dx) = 2 ln(sin(x)) csc(x) cos(x)#
Therefore #(dg)/(dx) = 2 ln(sin(x)) csc(x) cos(x)#. Substitute that back:
#(df)/(dx) = 2 ln(sin(x)) csc(x) cos(x) - (dh)/(dx) + (dp)/(dx)#
Now, on to #(dh)/(dx)#.
First using the product rule, we have that
#dh = d(3x ln(sin x)) = 3x * d(ln(six(x))) + ln(sin(x)) * d(3x)#
Then we need to find #d(3x)# and #d(ln(sin(x)))#. The first one is easy:
#(d(3x))/(dx) = 3 rarr d(3x) = 3 dx#
Putting that back:
#(dh) = 3x * d(ln(sin(x))) + 3 ln(sin(x)) dx#
As for #d(ln(sin(x)))#... well, this happens to be the same as the #dg_2# we calculated earlier, so:
#(dh) = 3x csc(x) cos(x) dx + 3 ln(sin(x)) dx#
#(dh)/(dx) = 3x csc(x) cos(x) + 3 ln(sin(x))#
Substituting this back:
#(df)/(dx) = 2 ln(sin(x)) csc(x) cos(x) - 3x csc(x) cos(x) - 3 ln(sin(x)) + (dp)/(dx)#
For #(dp)/(dx)#, we'll use the product rule again:
#dp = x^2 d(ln(cos^2 x^2)) + ln(cos^2 x^2) d(x^2)#
Let's do #d(x^2)#:
#(d(x^2))/(dx) = 2x rarr d(x^2) = 2x dx#
Put it back:
#dp = x^2 d(ln(cos^2 x^2)) + 2x ln(cos^2 x^2) dx#
Now we just need #d(ln(cos^2 x^2))#, which is more like #d(ln((cos(x^2))^2))#, and we'll label #dq#:
#dp = x^2 dq + 2x ln((cos(x^2))^2) dx#
Let's break down the composition of #dq#:
#q_1 (x) = x^2#
#q_2 (x) = cos(x)#
#q_3 (x) = x^2#
#q_4 (x) = ln(x)#
So #q(x) = q_4(q_3(q_2(q_1(x))))#. Yes, #q_1# and #q_3# are the same, but when peeling off the onion, we need different layers to be labeled differently.
#(dq_1)/(dx) = 2x rarr dq_1 = 2x dx#
#dq_2 = d(cos(q_1)) = -sin(q_1) dq_1 = -sin(x^2) 2x dx#
#(dq_2)/(dx) = -sin(x^2) 2x#
#dq_3 = d((q_2)^2) = 2q_2 dq_2 = 2cos(q_1) * -sin(x^2) 2x dx#
#= -4x sin(x^2) cos(x^2) dx#
#(dq_3)/(dx) = -4x sin(x^2) cos(x^2)#
#dq_4 = d(ln(q_3)) = (q_3)^-1 dq_3#
#= (q_2)^-2 * -4x sin(x^2) cos(x^2) dx#
#= (cos(q_1))^-2 * -4x sin(x^2) cos(x^2) dx#
#= (cos(x^2))^-2 * -4x sin(x^2) cos(x^2) dx#
#= -4x sin(x^2) cos(x^2) (cos(x^2))^-2 dx#
#= -4x sin(x^2) (cos(x^2))^-1 dx#
#= -4x sin(x^2) sec(x^2) dx#
#(dq_4)/(dx) = -4x sin(x^2) sec(x^2)#
So,
#(dq)/(dx) = -4x sin(x^2) sec(x^2) rarr dq = -4x sin(x^2) sec(x^2) dx#
Let's substitute this back into #(dp)/(dx)#:
#dp = x^2 * -4x sin(x^2) sec(x^2) dx + 2x ln((cos(x^2))^2) dx#
#dp = -4 x^3 sin(x^2) sec(x^2) dx + 2x ln((cos(x^2))^2) dx#
#(dp)/(dx) = -4 x^3 sin(x^2) sec(x^2) + 2x ln((cos(x^2))^2)#
And finally this into #(df)/(dx)#:
#(df)/(dx) = 2 ln(sin(x)) csc(x) cos(x) - 3x csc(x) cos(x) - 3 ln(sin(x)) - 4 x^3 sin(x^2) sec(x^2) + 2x ln((cos(x^2))^2)#
