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Multivariable Chain Rule – Proving an equation of partial derivatives – Exercise 6493

Exercise

Given the differentiable function (with parameters a,b)

w=f(x+at, y+bt)

Prove the equation

w'_t=aw'_x+bw'_y

Proof

Define

u=x+at

v=y+bt

We get the function

w=f(u,v)

And the internal functions

u(x,t)=x+at

v(y,t)=y+bt

We will use the chain rule to calculate the partial derivatives of w.

w'_t=f'_u\cdot u'_t+f'_v\cdot v'_t=

=f'_u\cdot a+f'_v\cdot b

w'_x=f'_u\cdot u'_x+f'_v\cdot v'_x=

=f'_u\cdot 1+f'_v\cdot 0=

=f'_u

w'_y=f'_u\cdot u'_y+f'_v\cdot v'_y=

=f'_u\cdot 0+f'_v\cdot 1=

=f'_v

We got the equations

w'_t=f'_u\cdot a+f'_v\cdot b

w'_x=f'_u

w'_y=f'_v

Hence, we get

w'_t=w'_x\cdot a+w'_y\cdot b

w'_t=aw'_x+bw'_y

As required.

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