How to find the derivative of a parametric curve
What is the derivative of a parametric curve?
Given a parametric curve where our function is defined by two equations, one for ???x??? and one for ???y???, and both of them in terms of a parameter ???t???,
???x=f(t)???
???y=g(t)???
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we calculate the derivative of the parametric curve using the formula
???\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}???
where ???dy/dx??? is the first derivative of the parametric curve, ???dx/dt??? is the derivative of ???x=f(t)??? and ???dy/dt??? is the derivative of ???y=g(t)???.
How do we find the derivative of a parametric curve?
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Using the formula to find the derivative of a parametric curve
Example
Find the derivative of the parametric curve.
???x=3t^4-6???
???y=2e^{4t}???
We’ll start by finding ???dy/dt??? and ???dx/dt???.
???y=2e^{4t}???
???\frac{dy}{dt}=8e^{4t}???
and
???x=3t^4-6???
???\frac{dx}{dt}=12t^3???
Plugging these into the derivative formula for ???dy/dx???, we get
???\frac{dy}{dx}=\frac{8e^{4t}}{12t^3}???
???\frac{dy}{dx}=\frac{2e^{4t}}{3t^3}???
