Hi there,

How would I solve this?

A tangent to the hyperbola y=1/(2x) , x>0 cuts the y- axis at the point y=2
Find the gradient of this tangent?

If I differentiate the hyperbola, I would get -1/(2x^2) and then I don't know how to go on about doing it.

Thanks

We are told that the line passes through y=2. When y=2, then x=0.

So, the line passes through (0,2), but we do not know where it is tangent to the curve.

We can use y-y_{1}=m(x-x_{1})

Differentiating, we find the slope is m=\frac{-1}{2x^{2}} and we know

y=\frac{1}{2x}.

So, let's plug everything in and solve for x.

\frac{1}{2x}-2=\frac{-1}{2x^{2}}(x-0)

Now, solve for x and plug that value into your derivative to find the slope at that point.

Then, if you want to go further, find the equation of the tangent line.