Answer:
[tex]y=2x-2[/tex]
Step-by-step explanation:
The equation of the tangent line can be found by using the formula [tex]y-y_1=m(x-x_1)[/tex] where [tex]m[/tex] is the slope and [tex](x_1,y_1)[/tex] are the coordinate points of the line.
Therefore, we'll need to find the slope of [tex]y=x^4-x^2[/tex] at the point where [tex]x=1[/tex] by taking its derivative and plugging [tex]x=1[/tex] into the derivative:
[tex]f(x)=x^4-x^2[/tex]
[tex]f'(x)=4x^3-2x[/tex] (Remember to use the Power Rule here!)
[tex]f'(1)=4(1)^3-2(1)[/tex]
[tex]f'(1)=4-2[/tex]
[tex]f'(1)=2[/tex] <-- Our slope here is 2
Now, evaluate [tex]f(1)[/tex] to get [tex]y_1[/tex]:
[tex]f(1)=1^4-1^2[/tex]
[tex]f(1)=0[/tex]
Therefore, the equation of the tangent line is:
[tex]y-0=2(x-1)[/tex]
[tex]y=2x-2[/tex]
See attached graph for a visual reference
