Derivative of a Parametic Function
How to find the derivative of a parametric function: 1 example and its solution.
Examplex = t3 - 2t, y = t2 + 1, dy/dx = ?
A parametric function is a function
that a parameter (t) connects the variables (x, y).
In this example,
x and y are not directly connected by a formula: like y = f(x).
But the parameter t connects both x and y.
Let's see how to find dy/dx
at t = 1.
x = t3 - 2t
So dx/dt = 3t2 - 2.
Derivative of a Polynomial
y = t2 + 1
So dy/dt = 2t1 + 0 = 2t.
It says to find dy/dx.
But you found dx/dt and dy/dt.
So change dy/dx to (dy/dt)/(dx/dt).
It's like dividing both of the numerator and the denominator by dt.
dy/dt = 2t
dx/dt = 3t2 - 2
Then (dy/dt)/(dx/dt) = 2t/(3t2 - 2).
Don't put dx/dt = 3t2 - 2 in the numerator.
It's easy to make this mistake.
It says to find dy/dx at t = 1.
So put t = 1 into dy/dx = 2t/(3t2 - 2).
Then [dy/dx]t = 1 = 2⋅1/(3⋅12 - 2).
Then you get 2.
So dy/dx = 2
at t = 1.