Explain like I'm five

[topaz spruce]

When differentiating, why is it when you differentiate something like 5xy you end up with

5x -> 5
y -> dy/dx.

GPT says 'because y depends on x'

Ok.. so

If its 5yx, would it be

5y -> 5
x -> dy/dx ?

What does it mean that it 'depends on the other variable' ?

And why, if x becomes dy/dx do you end up with

5y + 5x * dy/dx ?

[rain pollen]

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[stable atlas]

Because this is implicitly differentiating.

We know that y is a function of x, but we don’t know what the derivative of y with respect to x actually is.

So, we just denote it dy/dx.

However, we know the derivative of x with respect to x; it is 1.

So, using the product rule, we have 5y1+5xdy/dx=5y+5x*dy/dx.

[wintry crystal]

D/dx is an operator
Apply it to 5x and you get 5, since the function y = 5x changes at a rate of 5 (has gradient 5, I mean it’s literally a line with m equals 5)
Apply it to y and you get d/dx of y or dy/dx which you can see as a more ‘general’ case

[misty nacelle]

@topaz spruce

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