Differentiation with a bit of trig?

[akw]

Hi,

Please see the attached image.

The step that I don't understand is marked. Please could someone explain how this magic happens?

 

[MathLover1]

 

[akw]

Thanks But I meant the previous step. I should have been clearer.

 

[MathLover1]

Apply the constant multiple rule (d/dx)(c F(x)) with

and

=

apply the chain rule (d/dx(F(G(x)))=(d/du)(F(u))(d/dx)(G(x)):

=

apply the power rule (d/du)(u^n)=n*u^(n-1) with n=-1

=

=

Return to the old variable:

= will continue ( only 10 img can be in one post)

 

[MathLover1]

=

The derivative of a sum/difference is the sum/difference of derivatives:

=

The derivative of the cosine is (d/dx)(cos(x))=-sin(x):

=

Apply the constant multiple rule:


=

The derivative of the sine is (d/dx)(sin(x))=cos(x):

=

=

=

equal to zero

will be zero if one factor equal to zero, so let -sin(theta)+mu[k]cos(theta)=0

 

[MathLover1]

then

 

[akw]

Thanks so much. That's great.

What confuses me still, I suppose, is how one can "choose" to make one factor = 0, arbitrarily, if you see what I mean. It alarms my emotional brain, although I've checked, and I see that it does indeed yield a minimal T.

So why choose that factor to be 0? (i.e. why not choose the other?)

 

[MathLover1]

you want to find angle θ and it will be possible only if (dT)/(d(theta))=0

since

, you see that on right side is a product of two factors

and

then, you choose simpler factor which is

and equal it to zero