Question 826996
{{{y}}}{{{""=""}}}{{{x^4-x+1}}}
<pre>
The graph of that is gotten by plotting

(-2,19), (-1,3), (0,1), (1,1), (2,15) 

{{{drawing(2400/11,800,-3,3,-2,20,
circle(-2,19,0.15),circle(-2,19,0.13),circle(-2,19,0.11),circle(-2,19,0.09),circle(-2,19,0.07),circle(-2,19,0.05),circle(-2,19,0.03),circle(-2,19,0.01),
circle(-1,3,0.15),circle(-1,3,0.13),circle(-1,3,0.11),circle(-1,3,0.09),circle(-1,3,0.07),circle(-1,3,0.05),circle(-1,3,0.03),circle(-1,3,0.01),
circle(0,1,0.15),circle(0,1,0.13),circle(0,1,0.11),circle(0,1,0.09),circle(0,1,0.07),circle(0,1,0.05),circle(0,1,0.03),circle(0,1,0.01),

circle(1,1,0.15),circle(1,1,0.13),circle(1,1,0.11),circle(1,1,0.09),circle(1,1,0.07),circle(1,1,0.05),circle(1,1,0.03),circle(1,1,0.01),

circle(2,15,0.15),circle(2,15,0.13),circle(2,15,0.11),circle(2,15,0.09),circle(2,15,0.07),circle(2,15,0.05),circle(2,15,0.03),circle(2,15,0.01),

graph(2400/11,800,-3,3,-2,20,x^4-x+1)  )}}}

The slope of a tangent line at a point IS the derivative
at that point.  So we find {{{(dy)/(dx)}}}

{{{(dy)/(dx)}}}{{{""=""}}}{{{4x^3-1}}}

Evaluate that at x=1

{{{matrix(2,3,

(dy)/(dx),"|","",
    ""   ,"|",x=1)}}}{{{""=""}}}{{{4(1)^3-1}}}{{{""=""}}}{{{4-1}}}{{{""=""}}}{{{3}}}

Now we find the equation of the line with slope m=3 that
goes through to point where x=1.  We must find the y-coordinate,
by substituting x=1 in the original equation for the graph:

{{{y}}}{{{""=""}}}{{{x^4-x+1}}}
{{{y}}}{{{""=""}}}{{{1^4-1+1}}}
{{{y}}}{{{""=""}}}{{{1}}}

Use the point-slope formula:

y - y<sub>1</sub> = m(x - x<sub>1</sub>)
where (x<sub>1</sub>,y<sub>1</sub>) = (1,1)

y-1 = 3(x-1)
y-1 = 3x-3
  y = 3x-2    <---answer

Find the points on that tangent line (0,-2), (1,1), (2,4), (3,7)
and graph it as a check to make sure it looks like it's tangent 
to the curve:

{{{drawing(2400/11,800,-3,3,-2,20,
circle(-2,19,0.15),circle(-2,19,0.13),circle(-2,19,0.11),circle(-2,19,0.09),circle(-2,19,0.07),circle(-2,19,0.05),circle(-2,19,0.03),circle(-2,19,0.01),
circle(-1,3,0.15),circle(-1,3,0.13),circle(-1,3,0.11),circle(-1,3,0.09),circle(-1,3,0.07),circle(-1,3,0.05),circle(-1,3,0.03),circle(-1,3,0.01),
circle(0,1,0.15),circle(0,1,0.13),circle(0,1,0.11),circle(0,1,0.09),circle(0,1,0.07),circle(0,1,0.05),circle(0,1,0.03),circle(0,1,0.01),

circle(1,1,0.15),circle(1,1,0.13),circle(1,1,0.11),circle(1,1,0.09),circle(1,1,0.07),circle(1,1,0.05),circle(1,1,0.03),circle(1,1,0.01),

circle(2,15,0.15),circle(2,15,0.13),circle(2,15,0.11),circle(2,15,0.09),circle(2,15,0.07),circle(2,15,0.05),circle(2,15,0.03),circle(2,15,0.01),


circle(0,-2,0.15),circle(0,-2,0.13),circle(0,-2,0.11),circle(0,-2,0.09),circle(0,-2,0.07),circle(0,-2,0.05),circle(0,-2,0.03),circle(0,-2,0.01),

circle(1,1,0.15),circle(1,1,0.13),circle(1,1,0.11),circle(1,1,0.09),circle(1,1,0.07),circle(1,1,0.05),circle(1,1,0.03),circle(1,1,0.01),
circle(2,4,0.15),circle(2,4,0.13),circle(2,4,0.11),circle(2,4,0.09),circle(2,4,0.07),circle(2,4,0.05),circle(2,4,0.03),circle(2,4,0.01),


circle(3,7,0.15),circle(3,7,0.13),circle(3,7,0.11),circle(3,7,0.09),circle(3,7,0.07),circle(3,7,0.05),circle(3,7,0.03),circle(3,7,0.01),green(line(-5,-17,7,19)),graph(2400/11,800,-3,3,-2,20,x^4-x+1),locate(1,1,"(1,1)")  )}}}


Edwin</pre>