# What is the equation of the line tangent to f(x)=2x ^3+8x  at x=1?

Apr 1, 2016

$y = 14 x - 4$

#### Explanation:

The equation of the tangent line is: $y = k x + n$
where $k = f ' \left({x}_{0}\right) , {x}_{0} = 1$

So, $f ' \left(x\right) = \left(2 {x}^{3} + 8 x\right) ' = 6 {x}^{2} + 8$,

and hence $k = f ' \left(1\right) = 6 \cdot {1}^{2} + 8 = 14$

We have: $f \left({x}_{0}\right) = f \left(1\right) = 2 \cdot {1}^{3} + 8 \cdot 1 = 10$ and

$y \left({x}_{0}\right) = k {x}_{0} + n$

$10 = 14 \cdot 1 + n \implies n = - 4$

Finally, $y = 14 x - 4$