Differentiation

[Stpm]

Hello, does anyone know how to prove question 30 ( a)
I only know how to solve up to that step

 

[Dr.Peterson]

Hello, does anyone know how to prove question 30 ( a)
I only know how to solve up to that step

I'd first try to rewrite the numerator using an angle sum identity. After that, I'd differentiate.

 

[Stpm]

I’ll try that. Thanks! Also can I know what’s the first step for question 30( b ) ? I’m stuck solving that too

 

[Stpm]

I'd first try to rewrite the numerator using an angle sum identity. After that, I'd differentiate.

Sorry,I don’t know how to start. I’m supposed to write numerator as 3 sin ( x+x) + 4 cos (x+x) ?

 

[Dr.Peterson]

Sorry,I don’t know how to start. I’m supposed to write numerator as 3 sin ( x+x) + 4 cos (x+x) ?

No. You might have learned something about how to write [imath]a\sin(x) + b\cos(x)[/imath] as [imath]c\cos(x + d)[/imath]. But if not, then work backward! That's what I did.

You know the derivative is supposed to involve [imath]\cos(2x + \alpha)[/imath], where [imath]\tan\alpha=\frac{4}{3}[/imath], so try expanding that (or maybe [imath]\sin(2x + \alpha)[/imath], since it is the derivative) and see if you get [imath]3\sin(2x) + 4\cos(2x)[/imath].

When your task is to show that something is true and you are told what it is, it is not necessary to work forward, as if you didn't know the answer. This is like proving an identity: you can start with either side, and use the other side to guide the direction of your work. So, use the "hint".