I have a question about $\tan \theta = \frac{g'(t)}{f'(t)}$

Question

I saw an expression of tangent to be $$ \tan \theta = \frac{g'(t)}{f'(t)} $$. But I thought $tan \theta $ was just $\frac{y}{x}$ or $\frac{g(t)}{f(t)}$. Are the derivatives also fine?

Answer 1

Recall that, by definition of derivative, for a function $y=h(x)$ the angular coefficient for the tangent line at a point is given by

$$\tan \theta = m = \frac{dy}{dx}$$

and when the function is parametrized by $(x,y)=(f(t),g(t))$ for $f'(t) \neq 0$ we have

$$\tan \theta =\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{g'(t)}{f'(t)}$$

For example consider the parabola $y=x^2$ then the angular coefficient at $(1,1)$ is

$$m=\tan \theta = y'(1)=2$$

and by parametrization $x=f(t)=t$, $y=g(t)=t^2$ we have

$$m=\tan \theta = \frac{g'(1)}{f'(1)}=2$$