Q.Find $\dfrac{dy}{dx}$, if $x = e^{3t}$, $y = e^{(4t + 5)}$

Differentiate $x = e^{3t}$ and $y = e^{4t+5}$ with respect to $t$, then use $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ to get $\dfrac{4}{3}e^{\,t+5}$.

The curve is given in parametric form, so we differentiate each of $x$ and $y$ with respect to the parameter $t$ and divide.

Differentiate $x = e^{3t}$:

$\frac{dx}{dt} = e^{3t}\cdot \frac{d}{dt}(3t) = 3e^{3t}.$

Differentiate $y = e^{4t+5}$:

$\frac{dy}{dt} = e^{4t+5}\cdot \frac{d}{dt}(4t+5) = 4e^{4t+5}.$

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