Parametric function

Calculus Level 1

{ x = 2 t y = 3 t ⋅ t \large \begin{cases} x = 2t \\ y = 3t \cdot t \end{cases}

Given the system of equations above, find d y d x \dfrac {dy}{dx} .

2 t − 1 2t-1 4 t 4t x + 2 x+2 3 t 3t 2 t 2t x − t x-t

Mohammad Khaza by Mohammad Khaza , 20, Bangladesh

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2 solutions

Mohammad Khaza
Oct 4, 2017

at first,

d x d t = 2 \frac{dx}{dt}=2

d y d t = 6 t \frac{dy}{dt}=6t

now, d y d x = ( d y d t ) × ( d t d x ) \frac{dy}{dx}=(\frac{dy}{dt}) \times (\frac{dt}{dx}) = 6 t 2 = 3 t \frac{6t}{2}=3t

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Gandoff Tan
Jul 26, 2019

The long way around:

x = 2 t ⇒ t = 1 2 x x=2t\Rightarrow t=\frac12x

y = 3 ( 1 2 x ) 2 y=3{\left(\frac12x\right)}^2

y = 3 4 x 2 y=\frac34x^2

d y d x = 3 2 x \frac{dy}{dx}=\frac32x

d y d x = 3 2 ( 2 t ) \frac{dy}{dx}=\frac32\left(2t\right)

d y d x = 3 t \boxed{\frac{dy}{dx}=3t}

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