Question

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

Answer 1

`y=-4x-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 => nx^(n-1)` `leftarrow` `color(blue)(Note:"This is the process of the power rule."`

Knowing this we shall find our own derivative:

`x^4 => 4x^3`

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

`4x^3 => 4(-1)^3 => 4(-1) => -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 => (-1)^4 => 1`

We should now have:

`color(red)(x=-1)`
`color(blue)(y=1)`
`color(orange)(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:

`(y-color(blue)(y_1))=color(orange)(m)(x-color(red)(x_1))`

`=> (y-color(blue)(1))=color(orange)(-4)(x-color(red)((-1)))`

Simplify and solve:

`(y-1)=-4(x-(-1))`

`=> y-1=-4x-4`

`=> y=-4x-3`

Hope this helped!