Question

can the product rule be used to verify the chain rule? support your answer with example.

Answer 1

$\text{Yes, for a single variable}\\ \text{Let x, y be two given functions}\\ \text{Consider} \, f=(x+y)^2. \\ \text{By the chain rule, we have}\\ f'=2(x + y)(x' + y') = 2(xx'+ xy'+ yx'+ yy')\\ \text{Now, expand f to obtain} \, f=x^2+2xy + y^2\\ \text{Thus, we have}\\ f' =2xx'+2(xy)' +2yy' = 2(xx'+(xy)'+yy'))\\ \text{It follows immediately that}\\ (xy)'=xy' + yx' \\ \text{Thus, we see that equality holds}$