Find dy/dx: sin²y + cos xy = κ

Solution:

A derivative helps us to know the changing relationship between two variables. Consider the independent variable 'x' and the dependent variable 'y'.

The change in the value of the dependent variable with respect to the change in the value of the independent variable expression can be found using the derivative formula.

Given that

sin²y + cos x y = κ

Let us find the derivative on both sides with respect to x.

On differentiating with respect to x, we get

⇒ d/dx (sin²y) + d/dx(cos x y ) = d/dx (κ)

⇒ d/dx (sin²y) + d/dx(cos x y ) = 0-----(1)

Using chain rule of derivative, we obtain

⇒ d/dx (sin²y) = 2 sin y d/dx (sin y) = 2 siny cos y dy/dx-----(2)

⇒ d/dx(cos x y ) = - sin xy d/dx (xy)

= - sin xy [ y d/dx (x) + x dy/dx]

= - sin xy [ y . 1 + x dy/dx]

= - y sin xy - x sin xy dy/dx -----(3)

From equation 1, 2 and 3 we get

2 sin y cos y dy/dx - y sin xy - x sin xy dy/dx = 0

⇒ (2 sin y cos y - x sin xy) dy/dx = y sin xy

⇒ (sin 2y - x sin xy ) dy/dx = y sin xy 

Therefore,

dy/dx = (y sin xy) / (sin 2y - x sin xy )

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.3 Question 7

Find dy/dx: sin²y + cos x y = κ

Summary:

The derivative of  sin²y + cos x y = κ with respect to x is dy/dx = (y sin xy) / (sin 2y - x sin xy ) .A derivative helps us to know the changing relationship between two variables