Differentiation

The equation of a curve is $xy^2 + \ln(x + 2y) = 1$.

Find the gradient of the curve at the point where $x = 0$.

The parametric equations of a curve are

Find the equation of the tangent to the curve at the point $(e, 3)$. Give your answer in the form $y = mx + c$, where $m$ and $c$ are exact.

Find the exact coordinates of the stationary point of the curve with equation $y = 3x^3 \ln x^4$, for $x > 0$.

The diagram shows the graph of $y = e^{\sin 2x} \cos 4x$ for $0 \le x \le \frac{1}{4}\pi$, and its maximum point $M$.

Find the $x$-coordinate of $M$.

The parametric equations of a curve are

Obtain a simplified expression for $\frac{dy}{dx}$ in terms of $t$.

The equation of a curve is $x^2 \ln 2y - y \ln(2 + x^2) = \ln 6$.

Find the exact value of the gradient of the curve at the point $(2, 3)$. Give your answer in simplified form.