Answer to Question #100203 in Differential Equations for Beast

Question

Define derivatives for the following functions

a.) y= √3x^2+1/3t

b.) y= e^-x sin^2x

c.) y= sin2x / sin^2x

Answer 1

1)  y= √3x²+1/3t  

Thus,

dy/dx=d/dx(√3x²+1/3t)=2√3x+0=2√3x,

Assume t is constant here.

2)y= e^-x sin²x

dy/dx= e^-x d/dx( sin ²x)+sin²x d/dx(e^-x )

dy/dx=e^-x d/dx (1-cos2x)/2+sin²xd/dx(e^-x )

dy/dx=e^-x sin2x-sin^2x e^-x

3) y= sin2x / sin²x

y=2sin2x/(1-cos2x)

dy/dx={(1-cos2x) d/dx(2sin2x)-2sin2x d/dx(1-cos2x)}/(1-cos2x)²

dy/dx={4(1-cos2x)cos2x-4sin2x sin2x}/(1-cos2x)²

dy/dx={4cos2x-4cos²2x-4sin²2x}/(1-cos2x)²

dy/dx=(4cos2x-4)/(1-cos2x)²

dy/dx=-4/(1-cos2x)