Tangent equation

You also use the derivative to find the equation of the tangent to a curve at a specific point.

Example

Give the tangent equation of the function $f(x)=-x^2+3$ at $x=2$.

1. Find derivative

   Antiderivative: $f(x)=-x^2+3$
   Derivative: $f'(x)=-2x$

2. Calculate slope

   $f'(2)=-2\cdot2=-4$

3. Insert slope

   $t(x)=mx+n$
   
   $m=f'(2)=-4$
   $t(x)=-4x+n$

4. Calculate $n$ and insert

   The tangent passes through the point $P(2|f(2))$. In order to calculate $n$, the point is inserted into the equation
   $f(2)=-2^2+3=-1$
   => $P(\color{blue}{2}|\color{green}{-1})$
   
   $t(x)=-4x+n$
   $\color{green}{-1}=-4\cdot\color{blue}{2}+n$
   $-1=-8+n\quad|+8$
   $n=7$
   
   $t(x)=-4x+7$