How do you find #dy/dx# by implicit differentiation given #x=siny+cosy#?

Answer 1

# dy/dx = 1/(cos y -sin y) #

We have:

# x = sin y + cos y #

Method 1 - Use the Chain Rule

Differentiate wrt #y#;
# dx/dy = cos y -sin y #
By the chain rule; #dx/dy = 1/(dy/dx)#; so:
# dy/dx = 1/(cos y -sin y) #

Method 2 - Implicit Differentiation

Differentiate wrt #x#;
# \ \ \ \ \ \ \ \ 1 = d/dx(sin y + cos y) # # :. \ \ \ 1 = dy/dx*d/dy(sin y + cos y) # # :. \ \ \ 1 = dy/dx*(cos y -sin y) # # :. dy/dx = 1/(cos y -sin 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 2

To find ( \frac{dy}{dx} ) by implicit differentiation given ( x = \sin(y) + \cos(y) ), follow these steps:

  1. Differentiate both sides of the equation with respect to ( x ).
  2. Use the chain rule when differentiating terms involving ( y ).
  3. Solve for ( \frac{dy}{dx} ).

Differentiating ( x = \sin(y) + \cos(y) ) with respect to ( x ) yields:

[ \frac{dx}{dx} = \frac{d}{dx}(\sin(y) + \cos(y)) ]

[ 1 = \frac{d}{dy}(\sin(y)) \frac{dy}{dx} + \frac{d}{dy}(\cos(y)) \frac{dy}{dx} ]

[ 1 = \cos(y) \frac{dy}{dx} - \sin(y) \frac{dy}{dx} ]

[ \frac{dy}{dx} (\cos(y) - \sin(y)) = 1 ]

[ \frac{dy}{dx} = \frac{1}{\cos(y) - \sin(y)} ]

This is the derivative ( \frac{dy}{dx} ) in terms of ( 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