Let \(f(x)\) be a function differentiable at \(a\):
The derivative of \(f(x)\) at \(a\) is \(f'(a)\). Since the equation of a line is \(y=mx+c\), then we can define the tangent line like this:
\[y=f'(a)x + c\]
What about \(c\) though? Since the tangent line touches \(f(x)\) at \(a\), then:
\[\begin{gather} f(a)=f'(a) a + c \\ \therefore c = f(a) - f'(a) a \end{gather}\]
This means:
\[\begin{gather} y = f'(a)x + f(a) - f'(a)a \\ y = f(a) + f'(a)(x - a) \end{gather}\]
This is the equation of our tangent line.
