Q.If $y=e^{2x}$, find $\dfrac{d^2y}{dx^2}$.

The second order derivative $\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\!\left(\dfrac{dy}{dx}\right)$

measures how the slope of $y=f(x)$ is itself changing (physically, acceleration when $y$ is

position). For an explicit function, differentiate twice in succession. For an implicit

relation, differentiate the first-derivative expression again, substituting the known $dy/dx$ as

needed. For a parametric curve, the second derivative is

$\dfrac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}(dy/dx)}{dx/dt}$ -- differentiate the already-found

$dy/dx$ with respect to $t$ again and divide by $dx/dt$ once more -- never the invalid shortcut

$\dfrac{d^2y/dt^2}{d^2x/dt^2}$.