Question

How do you differentiate `y = 1/2 x + 1/4sin2x`?

Answer 1

`dy/dx = cos^2x`

Explanation:

This can be rewritten as follows using the identity `sin2theta = 2sinthetacostheta` and a little algebra.

`y = 1/2x + 1/4(2sinxcosx)`

`y = 1/2x + 1/2sinxcosx`

`y = 1/2(x + sinxcosx)`

`2y = x + sinxcosx`

Use implicit differentiation and the product rule to differentiate.

`2(dy/dx) = 1 + cosx(cosx) + sinx(-sinx)`

`2(dy/dx) = 1 + cos^2x - sin^2x`

Use the identity `cos^2theta + sin^2theta = 1 ->1 - sin^2theta = cos^2theta`:

`2(dy/dx) = cos^2x + cos^2x`

`2(dy/dx) = 2cos^2x`

`dy/dx = (2cos^2x)/2`

`dy/dx= cos^2x`

Hopefully this helps!