Question
Question: What is the derivative of \( y = \cos \left( {xy} \right) \) ?...
What is the derivative of y=cos(xy) ?
Solution
Hint : In the given problem, we are required to differentiate y=cos(xy) with respect to x. Since, y=cos(xy) is an implicit function, so we will have to differentiate the function y=cos(xy) with the implicit method of differentiation. So, differentiation of y=cos(xy) with respect to x will be done layer by layer using the chain rule of differentiation as in the given function, we cannot isolate the variables x and y.
Complete step by step solution:
Consider, y=cos(xy) .
Differentiating both sides of the equation with respect to x, we get,
⇒dxd(y)=dxd(cos(xy))
Using chain rule of differentiation, we will first differentiate cos(xy) with respect to xy and then differentiate xy with respect to x, we get,
⇒dxd(y)=d(xy)d(cos(xy))×dxd(xy)
⇒dxdy=−sin(xy)×dxd(xy)
Now, using product rule of differentiation, we have, dxd[f(x)g(x)]=f(x)dxdg(x)+g(x)dxdf(x) . So, we get,
⇒dxdy=−sin(xy)×[xdxdy+ydxd(x)]
Now, we know that the derivative of x with respect to x is 1 . So, we get,
⇒dxdy=−sin(xy)×[xdxdy+y]
Now, opening the brackets, we get,
⇒dxdy=−xsin(xy)dxdy−ysin(xy)
⇒dxdy+xsin(xy)dxdy=−ysin(xy)
⇒dxdy(1+xsin(xy))=−ysin(xy)
⇒dxdy=1+xsin(xy)−ysin(xy)
Therefore, we get the derivative of y=cos(xy) as 1+xsin(xy)−ysin(xy).
Note : Implicit functions are those functions that involve two variables and the two variables are not separable and cannot be isolated from each other. Hence, we have to follow a certain method to differentiate such functions and also apply the chain rule of differentiation. We must remember the simple derivatives of basic functions to solve such problems.