tan2:=x -> ln(2)*x+1;

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tan3:=x -> ln(3)*x+1;

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMiIiQiIiI5JEYyRjJGMkYyRiVGJUYl

Red curve : y=2^x Blue line: line tangent to 2^x at (0,1) Black line: line through (0,1) with slope 1 Notice: tangent line of 2^x at (0,1) has slope < 1 plot([2^x,tan2(x),x+1], x=-1..1,y=0..2, color=[red,blue,black], scaling=constrained);

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

Red curve : y=3^x Blue line: line tangent to 3^x at (0,1) Black line: line through (0,1) with slope 1 Notice: tangent line of 3^x at (0,1) has slope > 1 plot([3^x,tan3(x),x+1], x=-1..1,y=0..2, color=[red,blue,black], scaling=constrained);

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

Why isn't (e^x)' = xe^(x-1)? Look at the graphs: Red curve: e^x Blue curve: xe^(x-1) plot([exp(x), x*exp(x-1)], x=-3..1, color=[red,blue]);

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