Deriving trig functions?

Question

Im having some difficulty deriving a trig function using the product rule.
5x(sinx+cosx)
Should I even be using the product rule on this equation? I mean it does have a multiplying structure.

Answer 1

How is this any different than any other function? Yes... you have a trig function in there... so what? You could have any function of x in there. Use product rules, chain rules... it doesnt matter. Narrow down a single d/dx sin x and you make a simple substitution to cos x.

d/dx 5x(sinx+cosx) =

--- product rule ---
(sinx+cosx) d/dx 5x + 5x d/dx (sinx+cosx) =

(sin x + cos x) * 5 + 5x (cos x - sin x)

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Or, alternatively:

d/dx 5x(sinx+cosx) =

... distrubution
d/dx (5x sin x + 5x cos x) =

d/dx (5x sin x) + d/dx (5x cos x) =

... product rules
5x d/dx (sin x) + sin x d/dx 5x + 5x d/dx (cos x) + cos x d/dx 5x =

5x cos x + 5 sin x - 5x sin x + 5 cos x


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It doesnt matter how you do it. Just do it. Obey the rules, of course... but be creative. You will find the correct solution.

Answer 2

I think, yes, you need to use product rule. Because even you change it into 5xsinx + 5xcosx, they're still contains "product".

Suppose we say:
u = 5x
du/dx = 5
v = (sin x + cos x)
dv/dx = cos x - sin x

Product rule :
dy/dx = du/dx *v + u* dv/dx
=5(sinx + cos x) + 5x(cos x - sin x)
= .....

Answer 3

yes you should.use that formula F*G=F'*G+F*G'.
so it's like this 5x'*(sinx+cosx)+5x*(sinx+cosx)'=5(sinx+cosx)+5x(-cosx+sinx)=
5sinx+5cosx-5cosx''+5sinx''
i think so thats the final result.but anywhay this is the right path.try to do it ur self.good luck.

Answer 4

y = (5x) (sin x + cos x)
dy/dx = 5 (sin x + cos x) + (cos x - sin x) (5x)
dy/dx = (5 - 5x) sin x + (5 + 5x) cos x
dy/dx = 5(1 - x) sin x + 5(1 + x) cos x
dy/dx = 10 - 5x sin x + 5x cos x