Question

if x= 2 t³and y=3t² then the value of Dy/dx​

Answer 1

Given:

• x= 2 t³

• y = 3t²

To find :

• dy/dx

Solution:

Differentiate both x and y wr.t. t ,

For x :

$\implies \: x = 2 {t}^{3} \\ \\ \implies \: \ \frac{dx}{dt} = \frac{d}{dt} 2 {t}^{3 } \\ \\ \implies \: \frac{dx}{dt} = 6 {t}^{2}$

For y :

$\implies \: y = 3 {t}^{2} \\ \\ \implies \: \frac{dy}{dt} = \frac{d}{dt} 3 {t}^{2} \\ \\ \implies \: \frac{dy}{dt} = 6t$

Now :

$\star \: \frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } \\ \\ \star \: \frac{dy}{dx} = \frac{6t}{6 {t}^{2} } \\ \\ \ \star \: \frac{dy}{dx} = \frac{1}{t}$

Formula :

$\star \boxed{ \bold{ \red{\frac{d}{dx} {x}^{n} = n {x}^{n - 1} }}}$

Answer 2

Given ,

The two functions are

• x = 2(t)³
• y = 3(t)²

Differentiaing x wrt t , we get

$\tt \implies \frac{dx}{dt} = \frac{d \{2 {(t)}^{3} \}}{dt}$

$\tt \implies \frac{dx}{dt} = 6 {(t)}^{2}$

Similary , differentiaing y wrt t , we get

$\tt \implies \frac{dy}{dt} = \frac{d \{3 {(t)}^{2} \}}{dt}$

$\tt \implies \frac{dy}{dt} = 6t$

Now ,

dy/dx = dy/dt ÷ dx/dt

Thus ,

$\tt \implies \frac{dy}{dx} = \frac{6t}{6 {(t)}^{2} }$

$\tt \implies \frac{dy}{dx} = \frac{1}{t}$

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