Differentiation II

The chain rule brings the inner coefficient 5 out in front; the exponential itself is unchanged.

A standard result. Note it is undefined at $x = 0$, matching the domain of $\ln x$.

Derivative over function: the top is the derivative of the inside, $6x$.

Product rule: $(1)(\ln x) + (x)\left(\dfrac1x\right) = \ln x + 1$.

Differentiating gives $2x + 2y\dfrac{dy}{dx} = 0$. The radius to $(3,4)$ has gradient $\tfrac43$, and the tangent is perpendicular to it ✓

$\dfrac{dy}{dt} = 3\cos t$ and $\dfrac{dx}{dt} = -2\sin t$, so the ratio is $-\dfrac{3\cos t}{2\sin t}$.