How to find the derivative of a parametric curve

 
 
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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}???

 
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