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# Function of Function Rule

Linear Approximation and Differential of a function of several variables: 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 ŌłéW/Ōłéx , ŌłéW/Ōłéy . 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 ŌłÆ 2 y2 + 2xy and x (t) = cos t , y(t ) = sin t, t Ōłł[0, 2ŽĆ ] .

Solution

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

Then F ( x, y ) = cos2t ŌłÆ 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 (ŌłÆ sin t) ŌłÆ 4 sin t cos t + 2(ŌłÆsin2 t + cos2 t)

= ŌłÆ6 cos t sin t + 2(ŌłÆ sin2 t + cos2 t) .

On the other hand if we calculate = 2(cos t + sin t)(ŌłÆsin t ) + 2(cos t ŌłÆ 2 sin t )(cos t)

= ŌłÆ6 cos t sin t + 2(ŌłÆ sin2 t + cos2 t)

= dF /dt

Example 8.19

Let g ( x, y ) = x2 ŌłÆ yx + sin(x + y), x (t) = e3t , y(t ) = t2 , t Ōłł ŌäØ. 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 Ōłł ŌäØ. 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 ŌłéW/Ōłéx , ŌłéW/Ōłéy . If both variables x = x(s,t) and y = y(s,t), where s , t Ōłł ŌäØ, 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 cos╬Ė , and y = sin ╬Ė , r Ōēź 0 and ╬Ė Ōłł ŌäØ, (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 ŌłÆ s, y = r2 + 2s , r, s Ōłł ŌäØ. Find Solution

Here again we shall use the tree diagram to calculate Tags : Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives
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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Function of Function Rule | Mathematics