SOLUTION: Find x’ for the function x(t) defined implicitly below. Compute the equation of the tangent line at the indicated point. t ln x = xe^t-1 ; (t,x) = (0,1)

Algebra ->  Functions -> SOLUTION: Find x’ for the function x(t) defined implicitly below. Compute the equation of the tangent line at the indicated point. t ln x = xe^t-1 ; (t,x) = (0,1)       Log On


   

Question 1088245: Find x’ for the function x(t) defined implicitly below. Compute the equation of the tangent line at the indicated point.
t ln x = xe^t-1 ; (t,x) = (0,1)
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
t%2Aln%28x%29%22%22=%22%22x%2Ae%5Et-1+

Differentiating implicitly to find %22x%27%28t%29%22

t%2A%281%2Fx%29%2A%22x%27%22%2Bln%28x%29%22%22=%22%22x%2Ae%5Et%2Be%5Et%2A%22x%27%22

Clear the fraction by multiplying through by LCD=x

t%2A%22x%27%22%2Bx%2Aln%28x%29%22%22=%22%22x%5E2%2Ae%5Et%2Bx%2Ae%5Et%2A%22x%27%22

t%2A%22x%27%22-x%2Ae%5Et%2A%22x%27%22%22%22=%22%22x%5E2%2Ae%5Et-x%2Aln%28x%29

%28t-x%2Ae%5Et%29%2A%22x%27%22%22%22=%22%22x%5E2%2Ae%5Et-x%2Aln%28x%29

%22x%27%22=%28x%5E2%2Ae%5Et-x%2Aln%28x%29%29%2F%28t-x%2Ae%5Et%29

When y is the dependent variable and x is the independent
variable, the point-slope equation of the line through
(x1,y1) with slope m is y-y%5B1%5D%22%22=%22%22m%28x-x%5B1%5D%29.

But in this case, x is the dependent variable and t is the independent
variable, so the equation of the line through (t1,x1)
with slope m is x-x%5B1%5D%22%22=%22%22m%28t-t%5B1%5D%29.

(t1,x1) = (0,1)

And the slope m is x' evaluated at this point:

%22x%27%280%2C1%29%22=%281%5E2%2Ae%5E0-1%2Aln%281%29%29%2F%280-1%2Ae%5E0%29

%22x%27%280%2C1%29%22%22%22=%22%22%281%2A1-1%2A0%29%2F%28-1%2A1%29

%22x%27%280%2C1%29%22%22%22=%22%22-1
 
So m%22%22=%22%22-1

So substituting m = -1, and (t1,x1) = (0,1)

x-x%5B1%5D%22%22=%22%22m%28t-t%5B1%5D%29

x-1%22%22=%22%22-1%28t-0%29

x-1%22%22=%22%22-t

x%22%22=%22%22-t%2B1

Edwin