If {{MATH_8}} and {{MATH_9}} are differentiable, then {{MATH_10}} {{CMD_11}} To start, we need some way to understand the function {{MATH_12}} One interpretation of multiplication is it is the area of a {{MATH_13}} rectangle: {{CMD_14}} {{ENV_15}} {{CMD_16}} To understand the derivative of the product, we must understand how the area, {{MATH_17}}, changes as {{MATH_18}} changes. If we change the inputs of {{MATH_19}} and {{MATH_20}} by {{MATH_21}}, then the size of the rectangle changes: {{CMD_22}} {{ENV_23}} {{CMD_24}} However, we know from our previous work that {{ENV_25}} so now our picture becomes: {{CMD_26}} {{ENV_27}} {{CMD_28}} Note, if we think of {{MATH_29}}, then we can also label our picture as follows: {{CMD_30}} {{ENV_31}} {{CMD_32}} Finally, from the pictures above and recalling that {{ENV_33}} we see that {{ENV_34}} Dividing both sides by {{MATH_35}} we see {{MATH_36}} and letting {{MATH_37}} go to zero we see {{MATH_38}} {{CMD_39}} {{CMD_40}}