Sometimes you want to find the common tangent line of two functions. The first thing that comes to mind to a person that is learning basic calculus is that you should equal the derivatives of those functions. Nevertheless, this way to resolve a problem like this is inaccurate. I saw some questions in the site that show how to resolve this type of problem. However, no one explained why this way is wrong.
Both functions have the same slope! Why is it wrong to do it that way? What are we missing?
Best Answer
Consider the functions $f(x) = x^2$ and $g(x) = x^2 + 1$. They both have the same derivative at 0, $f'(0) = g'(0) = 0$, but they have different tangent lines $y=0$ and $y=1$.
What really needs to happen for two differentiable functions $f$ and $g$ to have a same tangent line is for there to be two values $a$ and $b$, such that the tangent lines to $f$ and $g$ respectively have both the same slope and the same y intercept.
For example, $f(x) = x^2$ and $g(x) = (x-1)^2$ both have the tangent line $y=0$.