# Differentiation with a bit of trig?

Discussion in 'Differentiation and Integration' started by akw, Sep 12, 2022.

1. ### akw

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Hi,

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

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akw, Sep 12, 2022
2. ### MathLover1

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MathLover1, Sep 20, 2022
3. ### akw

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

akw, Sep 21, 2022
4. ### MathLover1

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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

=

=

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

MathLover1, Sep 21, 2022
5. ### MathLover1

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=
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

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Last edited: Sep 21, 2022
MathLover1, Sep 21, 2022
6. ### MathLover1

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then

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MathLover1, Sep 21, 2022
7. ### akw

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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?)

akw, Sep 21, 2022
8. ### MathLover1

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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