# For f(x)=x^4 what is the equation of the tangent line at x=-1?

Apr 5, 2018

$y = - 4 x - 3$

#### Explanation:

To find the equation of a tangent line, we must first find the derivative of the initial function.

The process of finding the derivative, in this case, can be simply put as:

${x}^{n} \implies n {x}^{n - 1}$ $\leftarrow$ color(blue)(Note:"This is the process of the power rule."

Knowing this we shall find our own derivative:

${x}^{4} \implies 4 {x}^{3}$

Now that we have our derivative, we can plug in the $- 1$ to give us the slope of the tangent line:

$4 {x}^{3} \implies 4 {\left(- 1\right)}^{3} \implies 4 \left(- 1\right) \implies - 4$

To continue on with finding the equation, we do need a $y$ value.

All we do here is plug in the $- 1$ into the original equation, and that will give us the value we need:

${x}^{4} \implies {\left(- 1\right)}^{4} \implies 1$

We should now have:

$\textcolor{red}{x = - 1}$
$\textcolor{b l u e}{y = 1}$
$\textcolor{\mathmr{and} a n \ge}{m = - 4}$

Having both of the necessary values, and the slope, we can use the point-slope form equation to find our equation for the tangent line:

$\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{\mathmr{and} a n \ge}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

$\implies \left(y - \textcolor{b l u e}{1}\right) = \textcolor{\mathmr{and} a n \ge}{- 4} \left(x - \textcolor{red}{\left(- 1\right)}\right)$

Simplify and solve:

$\left(y - 1\right) = - 4 \left(x - \left(- 1\right)\right)$

$\implies y - 1 = - 4 x - 4$

$\implies y = - 4 x - 3$

Hope this helped!