Find dydx, if x=2t² and y=6t⁶

The correct option is "9t⁴".

To find the derivative of y with respect to x, denoted as $\frac{dy}{dx}$, when both variables are expressed in terms of a parameter $t$ (parametric equations), we use the chain rule. The parametric differentiation formula is given by:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Let us first find the derivative of $x$ with respect to $t$:
Given: $x=2t^2$
Using the power rule of differentiation ($\frac{d}{dt}\left(t^n\right)=n{t}^{n-1}$), we get:

$\frac{dx}{dt}=2·2t^{2-1}=4t$

Next, we find the derivative of $y$ with respect to $t$:
Given: $y=6t^6$
Applying the power rule again:

$\frac{dy}{dt}=6·6t^{6-1}=36t^5$

Now, we substitute these derivatives into the parametric differentiation formula:

$\frac{dy}{dx}=\frac{36t^5}{4t}$

Simplifying the expression by dividing the coefficients and subtracting the exponents of $t$ (since $\frac{t^5}{t}=t^{5-1}=t^4$):

$\frac{dy}{dx}=9t^4$