# For f(x) =(1+2x)^2, what is the equation of the line tangent to x =1 ?

Apr 21, 2018

The equation for the tangent line is

$y = 12 x - 3$.

#### Explanation:

If

$f \left(x\right) = {\left(1 + 2 x\right)}^{2}$, then

$f ' \left(x\right) = 4 \left(1 + 2 x\right)$,

$f \left(1\right) = {\left[1 + 2 \left(1\right)\right]}^{2} = 9$, and

$f ' \left(1\right) = 4 \left(1 + 2\right) = 12$

The equation for the tangent line is

$y = 12 x + b$.

We can use the known point $\left(1 , 9\right)$ to find $b$.

$9 = 12 \left(1\right) + b$

$b = - 3$

The equation for the tangent line is

$y = 12 x - 3$.