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

1 Answer
Apr 5, 2018

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!