Question

Find the equation of the tangent line to the curve at the given point

x=cos(y); (1/2, 1/3pi)

I know for sure its

y- 1/3pi = ? (x-1/2)

Because its x= instead of y= would the deri be different?

Answer 1

Yes the derivative would be a little different.
We're looking to solve for \(\Large \dfrac{dy}{dx}\) or \(\large y'\).

The slope of our tangent line is given by \(\Large y'(1/2)=m\)

Answer 2

Woops, lemme restate that.
In this particular case, our derivative will be a function of x AND y.
So the slope of our tangent line will be given by:\[\Large y'(1/2,\quad 1/3\pi)=m\]

Answer 3

So we start by taking the derivative, with respect to x,\[\Large 1=(-\sin y)y'\]

Answer 4

\[\Large y'=\frac{-1}{\sin y}\]Evaluating the derivative function at x=1/2, y=pi/3 gives us,\[\Large m=\frac{-1}{\sin(\pi/3)}\]