Revision of Partial Differentiation

We revise partial differentiation by looking at a few examples.

Example 1. .9

1. 
   
   $$u = x^{2}\sin t,$$
   
   
   
   
   $$\Rightarrow \qquad {\partial u \over \partial x} = 2x \sin t, \qquad {\partial u \over \partial t} = x^{2} \cos t.$$
   
   

2. 
   
   $$u = \left (x^{2} + t^{2}\right )^{2} + e^{x},$$
   
   
   
   
   $$\Rightarrow \qquad {\partial u \over \partial x} = 2\left (... ...partial u \over \partial t} = 4t\left (x^{2} + t^{2}\right ).$$
   
   

3. 
   
   $$u = \tan^{-1}\left ({x\over t}\right ),$$
   
   
   
   
   $$\Rightarrow \qquad {\partial u \over \partial x} = {1\over 1 + x^{2}/t^{2}}.{1\over t} = {t\over x^{2} + t^{2}},$$
   
   
   
   
   $$\hbox{and }\qquad {\partial u \over \partial t} = {1\over 1 + x^{2}/t^{2}}.{-x\over t^{2}} = {-x\over x^{2} + t^{2}}.$$
   
   

If $u = u(r,s)$, $r = r(x,t)$ and $s = s(x,t)$, then the chain rule gives

$${\partial u\over \partial x} = {\partial u\over \partial r}... ...x} + {\partial u\over \partial s}{\partial s\over \partial x}.$$

$${\partial u\over \partial t} = {\partial u\over \partial r}... ...t} + {\partial u\over \partial s}{\partial s\over \partial t}.$$

Example 1. .10 $u = \sin (r + s^{2})$ and $r = te^{x}$ and $s = x+t$. Hence,

$${\partial r \over \partial x} = e^{x}t, \qquad {\partial s ... ...partial t} = e^{x}, \qquad {\partial s\over \partial t} = 1.$$

and so

$${\partial u\over \partial x} = \cos (r + s^{2}).te^{x} + \c... ... e^{x} + (x+t)^{2}\right ]\left \{t e^{x} + 2(x+t)\right \},$$

$${\partial u\over \partial t} = \cos (r + s^{2}).e^{x} + \co... ...t e^{x} + (x+t)^{2}\right ]\left \{ e^{x} + 2(x+t)\right \}.$$

This could have been obtained directly by substituting for $r$ and $s$ before differentiating so that $u = \sin \left ( t e^{x} + (x + t)^{2}\right )$. Thus, for example, we get

$${\partial u\over \partial t} = \cos \left [t e^{x} + (x+t)^{2}\right ]\left \{ e^{x} + 2(x+t)\right \},$$

directly. However, the chain rule is more powerful and will be used in the theory of the second order wave equation.