Dividing natural logs with derivatives? Help.

Note: It's not In with a large i, it's ln with a small L (logarithm)

What kind of function is it? f(x,y) multivariate? Then what kind of derivative are you trying to find? ∂f(x,y)/∂x, or ∂f(x,y)/∂y?
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I'm not entirely clear what the screenshot is showing. Each term is differentiated individually

∂[100+30xln(x)+20xln(y)]/∂x=0+[30x(1/x)+30(1)ln(x)]+20(1)ln(y)=0+30+30ln(x)+20ln(y)

The first term is 100, which is a constant so its derivative is 0.

The second term is 30xln(x) whicoih you differentiate using the product rule.

The third term is a function of x and y but the partial wrt x means you treat y as a constant so 20y is a constant and you differentiate x.

When you compute the partial wrt y then you change whicih variable is treated as a constant.

∂[100+30xln(x)+20xln(y)]/∂y=0+0+20x(1/y)