পোলার ও পরামিতিক স্থানাংকে ঢাল

IN PROGRESS

Let's review what a parametric function is.

Good! Now, let's say we graph a parametric function, and we want to find the slope of the tangent line to the graph at some point.

Well, what's the slope? We know we want $\dfrac{dy}{dx}$. How can we get that if all we have is $t$ as the variable?

We can use the chain rule! By the chain rule, we have that $$\dfrac{dy}{dt} = \dfrac{dy}{dx}\dfrac{dx}{dt}$$

If $\dfrac{dx}{dt} \not = 0$, then dividing by $\dfrac{dx}{dt}$ gives us $$\boxed{\dfrac{dy}{dx} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}}$$

This is known as the chain rule for parametric equations.

POLAR GRAPHS WILL BE ADDED SOON

By the chain rule or implicit differentiation, the slope of a parametric curve $(x(t),y(t))$ at time $t=a$, at the point $(x(a),y(a))$, is $$\frac{dy}{dx}|_{t=a} = \frac{dy/dt}{dx/dt}|_{t=a} = \frac{y'(a)}{x'(a)}.$$

As a special case, the slope of a polar curve $r = f(\theta)$ at angle $\theta = c$, at the point $(f(c)\cos{c},f(c)\sin{c})$, is $$\frac{\frac{d}{d\theta}(f(\theta)\sin\theta)|_{\theta=c}}{\frac{d}{d\theta}(f(\theta)\cos\theta)|_{\theta=c}} = \frac{f(c)\cos{c} + f'(c)\sin{c}}{-f(c)\sin{c} + f'(c)\cos{c}} .$$