Question

How do you use implicit differentiation to find the slope of the line tangent to `y+ lnxy=4` at (.25, 4)?

Answer 1

`dy/dx=-16/5` =`[3.2`] [The gradient]

Explanation:

`y+lnxy=4`, so by the theory of logs, `y+ lnx+lny=4`

Differentiating both sides of the equation implicitly with respect to x,....... `dy/dx+1/x+1/ydy/dx=0`, Therefore,

`dy/dx[1+1/y]=-1/x`...... so `dy/dx=-y/[x[y+1]]` and substituting for `x=0.25 and y=4` will give the above result for the gradient.