. . Function of Function Rule Let F be a function of two variables x y . Sometimes these variables may be functions of a single variable having same domain.
In this case, the function F ultimately depends only on one variable. So we should be able to treat this F as a function of single variable and study about dF dt . In fact, this is not a coincidence, it can be proved that Theorem . Suppose that W x y ( , ) is a function of two variables x y having partial derivatives ∂ W W .
If both the variables x y are differentiable functions of a single variable t , then W is a differentiable function of t and dW dt W dt W dt = ∂ + ∂ ...( ) Let us consider an example illustrating the above theorem. Example . Verify the above theorem for F x y ( , ) = and x t t y t t t cos , ( ) sin , [ , ] ∈ π . Let F ( x , y ) = x – y + xy and x ( t ) = cos t , y ( t ) = sin t .
Then F x y cos sin and thus F has becomes a function of one variable t . So by using chain rule, we see that dF dt = cos ( sin ) sin cos ( sin = − cos sin ( sin t . On the other hand if we calculate + ∂ dt dt = ( y dx dt y dy dt = (cos sin )( sin ) (cos sin )(cos ) = − cos sin ( sin = dF dt . and - - Differentials and Partial Derivatives Example .
Let g x y yx x t y t sin( ), . Find dg dt . We shall follow the tree diagram to calculate dg dt . So first we need to find ∂ g g dt and dy dt .
Now ∂ = −+ g g dt e t cos( ), cos( ), and dy dt = . Thus dg dt = ∂ + ∂ g dt g dt = + −+