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

Question

Emilia Aguilar · Accepted Answer

#y=-4x-3#

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!

Axel Alvarez · Answer

undefined The equation of the tangent line at x=-1 for f(x)=x^4 is y=-4x-4.