Math Derivative applications Tangent equation

# Tangent equation

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

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### Remember

The equation of the tangent line at $x$ is:
$t(x)=mx+n$
The slope of the tangent can be determined with the derivative at $x$:
$m=f'(x)$
i

### Method

1. Find derivative
2. Calculate slope
3. Insert slope
4. Calculate $n$ and insert

### 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
$f(2)=-2^2+3=-1$
=> $P(\color{blue}{2}|\color{green}{-1})$

$t(x)=-4x+n$
$\color{green}{-1}=-4\cdot\color{blue}{2}+n$
$-1=-8+n\quad|+8$
$n=7$

$t(x)=-4x+7$