Differentiation

Find $\frac{dy}{dx}$.

Hence find the coordinates of the stationary points of the curve.

Show that $\frac{dy}{dx} = \frac{-2e^{2x}y - 4e^{3x}}{e^{2x} + y^2}$.

Show that the curve has no stationary points.

Show that $\frac{dy}{dx}$ can be expressed in the form $c(3t + 4)$ and state the value of the constant $c$.

It is given that the gradient of the curve at the point $(a, \ln 100)$ is $m$.

Find the values of $a$ and $m$.

State whether the curve represents a decreasing function or an increasing function or neither. Give a reason for your answer.

Show that $\frac{dy}{dx} = 6\cos^5\theta - 5\cos^3\theta$.

Find the equation of the normal to the curve at the point where it crosses the $x$-axis. Give your answer in the form $y = mx + c$, where $m$ and $c$ are exact constants.

Find an expression in terms of $x$ and $y$ for $\frac{dy}{dx}$ and hence find the gradient of the curve at the point for which $y = 0$.

Show that there is no point on the curve at which the tangent is parallel to the $y$-axis.