Function of Function Rule

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 8.2

Suppose that W ( x, y) is a function of two variables x , y having partial derivatives ∂W/∂x , ∂W/∂y . If both the variables x , y are differentiable functions of a single variable t , then W is a differentiable function of t and

Let us consider an example illustrating the above theorem.

Example 8.18

Verify the above theorem for F ( x, y ) = x² − 2 y² + 2xy and x (t) = cos t , y(t ) = sin t, t ∈[0, 2π ] .

Solution

Let F(x,y) = x² – 2y² + 2xy and x(t) = cost, y(t) = sint.

Then F ( x, y ) = cos²t − 2 sin²t + 2 cos t sin t and thus F has becomes a function of one

variable t . So by using chain rule, we see that

dF/dt  = 2 cos t (− sin t) − 4 sin t cos t + 2(−sin² t + cos² t)

 = −6 cos t sin t + 2(− sin² t + cos² t) .

On the other hand if we calculate

 = 2(cos t + sin t)(−sin t ) + 2(cos t − 2 sin t )(cos t)

 = −6 cos t sin t + 2(− sin² t + cos² t)

= dF /dt

Example 8.19

Let g ( x, y ) = x² − yx + sin(x + y), x (t) = e^3t , y(t ) = t² , t ∈ ℝ. Find dg/dt.

Solution

We shall follow the tree diagram to calculate.

Also, some times our W ( x, y) will be such that x = x(s , t) , and y = y(s , t) where s , t ∈ ℝ. Then W can be considered as a function that depends on s and t . If x , y both have partial derivatives with respect to s , t and W has partial derivatives with respect to x and y , then we can calculate the partial derivatives of W with respect to s and t using the following theorem.

Theorem 8.3

Suppose that W ( x, y) is a function of two variables x , y having partial derivatives ∂W/∂x , ∂W/∂y . If both variables x = x(s,t) and y = y(s,t), where s , t ∈ ℝ, have partial derivatives with respect to both s and t, then

We omit the proof. The above theorem is very useful. For instance, consider the situation in which x = r cosθ , and y = sin θ , r ≥ 0 and θ ∈ ℝ, (change from cartesian co-ordinate to polar co-ordinate system). The above theorem can be generalized for functions having n number of variables.

Let us consider an example.

Example 8.20

Let g ( x, y ) = 2 y + x ² , x = 2r − s, y = r² + 2s , r, s ∈ ℝ. Find

Solution

Here again we shall use the tree diagram to calculate