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**Gusbob** If I recall correctly, this is how high schoolers are taught: to find a tangent line at the point $\displaystyle (a,f(a))$, you take $\displaystyle L=\displaystyle{\lim_{\Delta x\to 0}} \frac{f(a+\Delta x)-f(a)}{\Delta x}$. This gives you the slope of the tangent line. Since the tangent line passes through the point $\displaystyle (a,f(a))$, the point-slope formula gives $\displaystyle y-f(a)=L(x-a)$, which can be rearranged as $\displaystyle y=f(a)+L(x-a)$. Here's the kicker: $\displaystyle f'(a)=L=\displaystyle{\lim_{\Delta x\to 0}} \frac{f(a+\Delta x)-f(a)}{\Delta x}$ __by definition__, so you actually do get $\displaystyle y=f(a)+f'(a)(x-a)$.