How do you implicitly differentiate #y /sin( 2x) = x cos(2x-y)#?

Answer 1

#y'=(sin2xcos(2x-y)+2xcos(4x-y))/(1-xsin2xsin(2x-y))#

Let us rewrite the given eqn. as, #y=xsin2xcos(2x-y).....(1)#.
Keeping in mind that, #(uvw)'=u'vw+uv'w+uvw'#,
we diff. both sides of #(1)# w.r.t. #x#, to get,
#y'=1*sin2xcos(2x-y)+x(2cos2x)cos(2x-y)+xsin2x(cos(2x-y))'#

Here, by Chain Rule,

#(cos(2x-y))'=(-sin(2x-y))(2x-y)'=(-sin(2x-y))(2-y')=(y'-2)(sin(2x-y)#

Hence,

#y'=sin2xcos(2x-y)+2xcos2xcos(2x-y)+x(y'-2)sin2xsin(2x-y)#
#=sin2xcos(2x-y)+2xcos2xcos(2x-y)#
#+xy'sin2xsin(2x-y)-2xsin2xsin(2x-y)#
#:. y'(1-xsin2xsin(2x-y))=sin2xcos(2x-y)+2x{cos2xcos(2x-y)-sin2xsin(2x-y)}#
#=sin2xcos(2x-y)+2x{cos(2x+(2x-y))}#
#=sin2xcos(2x-y)+2xcos(4x-y)#

Therefore,

#y'=(sin2xcos(2x-y)+2xcos(4x-y))/(1-xsin2xsin(2x-y))#

Enjoy Maths.!

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To implicitly differentiate the equation ( \frac{y}{\sin(2x)} = x \cos(2x-y) ), follow these steps:

  1. Differentiate each term of the equation with respect to ( x ).
  2. Use the quotient rule to differentiate ( \frac{y}{\sin(2x)} ) and the chain rule to differentiate ( \cos(2x-y) ).
  3. Isolate ( \frac{dy}{dx} ) by solving for it.

After applying these steps, the result of the implicit differentiation will yield ( \frac{dy}{dx} ) in terms of ( x ) and ( y ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7